1915:
2547:
453:
3252:
2561:
2913:
2088:
29:
1159:
201:
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Several other shapes can be defined from a set of points in a similar way to the convex hull, as the minimal superset with some property, the intersection of all shapes containing the points from a given family of shapes, or the union of all combinations of points for a certain type of combination.
904:
In two dimensions, the convex hull is sometimes partitioned into two parts, the upper hull and the lower hull, stretching between the leftmost and rightmost points of the hull. More generally, for convex hulls in any dimension, one can partition the boundary of the hull into upward-facing points
1308:
of a convex set is a point in the set that does not lie on any open line segment between any other two points of the same set. For a convex hull, every extreme point must be part of the given set, because otherwise it cannot be formed as a convex combination of given points. According to the
3332:, the term "convex hull" had become standard; Dines adds that he finds the term unfortunate, because the colloquial meaning of the word "hull" would suggest that it refers to the surface of a shape, whereas the convex hull includes the interior and not just the surface.
2819:
of a finite point set give a nested family of (non-convex) geometric objects describing the shape of a point set at different levels of detail. Each of alpha shape is the union of some of the features of the
Delaunay triangulation, selected by comparing their
3210:
can make the minimum convex polygon excessively large, which has motivated relaxed approaches that contain only a subset of the observations, for instance by choosing one of the convex layers that is close to a target percentage of the samples, or in the
3287:
of a material, only those measurements on the lower convex hull will be stable. When removing a point from the hull and then calculating its distance to the hull, its distance to the new hull represents the degree of stability of the phase.
1317:) is the convex hull of its extreme points. However, this may not be true for convex sets that are not compact; for instance, the whole Euclidean plane and the open unit ball are both convex, but neither one has any extreme points.
1262:
2520:
structures can keep track of the convex hull for points moving continuously. The construction of convex hulls also serves as a tool, a building block for a number of other computational-geometric algorithms such as the
905:(points for which an upward ray is disjoint from the hull), downward-facing points, and extreme points. For three-dimensional hulls, the upward-facing and downward-facing parts of the boundary form topological disks.
3000:, points that do not belong to the hyperbolic space itself but lie on the boundary of a model of that space. The boundaries of convex hulls of ideal points of three-dimensional hyperbolic space are analogous to
2702:
or rectilinear convex hull is the intersection of all orthogonally convex and connected supersets, where a set is orthogonally convex if it contains all axis-parallel segments between pairs of its points.
2828:
of a point set are a nested family of convex polygons, the outermost of which is the convex hull, with the inner layers constructed recursively from the points that are not vertices of the convex hull.
1590:
commute with each other, in the sense that the
Minkowski sum of convex hulls of sets gives the same result as the convex hull of the Minkowski sum of the same sets. This provides a step towards the
2235:, a number of algorithms are known for computing the convex hull for a finite set of points and for other geometric objects. Computing the convex hull means constructing an unambiguous, efficient
1178:
is always itself open, and the convex hull of a compact set is always itself compact. However, there exist closed sets for which the convex hull is not closed. For instance, the closed set
1606:
operation to constructing the convex hull of a set of points is constructing the intersection of a family of closed halfspaces that all contain the origin (or any other designated point).
2504:
1878:
2016:
1666:
7007:
5586:
Kim, Sooran; Kim, Kyoo; Koo, Jahyun; Lee, Hoonkyung; Min, Byung Il; Kim, Duck Young (December 2019), "Pressure-induced phase transitions and superconductivity in magnesium carbides",
1037:), then it equals the closed convex hull. However, an intersection of closed half-spaces is itself closed, so when a convex hull is not closed it cannot be represented in this way.
4677:
3175:
is that it lies within the convex hull of its control points. This so-called "convex hull property" can be used, for instance, in quickly detecting intersections of these curves.
2900:
helps find their crossings, and convex hulls are part of the measurement of boat hulls. And in the study of animal behavior, convex hulls are used in a standard definition of the
2813:
2775:
1729:
2077:
1459:
3105:
of solutions to a combinatorial problem. If the facets of these polytopes can be found, describing the polytopes as intersections of halfspaces, then algorithms based on
2645:
intersects the object. Equivalently it is the intersection of the (non-convex) cones generated by the outline of the object with respect to each viewpoint. It is used in
172:. Convex hulls have wide applications in mathematics, statistics, combinatorial optimization, economics, geometric modeling, and ethology. Related structures include the
2420:
2378:
147:
2678:
1578:, the shelling antimatroid of the point set. Every antimatroid can be represented in this way by convex hulls of points in a Euclidean space of high-enough dimension.
3074:
form a nested family of convex sets, with the convex hull outermost, and the bagplot also displays another polygon from this nested family, the contour of 50% depth.
2036:
1976:
2454:
2209:. The definition can be extended to the convex hull of a set of functions (obtained from the convex hull of the union of their epigraphs, or equivalently from their
5568:
Kernohan, Brian J.; Gitzen, Robert A.; Millspaugh, Joshua J. (2001), "Analysis of animal space use and movements", in
Millspaugh, Joshua; Marzluff, John M. (eds.),
5411:
Hautier, Geoffroy (2014), "Data mining approaches to high-throughput crystal structure and compound prediction", in Atahan-Evrenk, Sule; Aspuru-Guzik, Alan (eds.),
846:
1696:
794:
417:
of the points encloses them with arbitrarily small surface area, smaller than the surface area of the convex hull. However, in higher dimensions, variants of the
1148:
1101:
2643:
2623:
2340:
2316:
2296:
2276:
2207:
2183:
2159:
1898:
1828:
1808:
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1753:
1568:
1548:
1528:
1499:
1479:
1433:
1413:
1384:
1364:
1121:
1078:
1058:
1027:
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983:
959:
894:
874:
814:
768:
748:
728:
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681:
661:
641:
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558:
538:
518:
498:
478:
411:
391:
367:
332:
306:
280:
258:
234:
7000:
6073:
Nilsen, Erlend B.; Pedersen, Simen; Linnell, John D. C. (2008), "Can minimum convex polygon home ranges be used to draw biologically meaningful conclusions?",
2992:
The definitions of a convex set as containing line segments between its points, and of a convex hull as the intersection of all convex supersets, apply to
1735:, and (by the Krein–Milman theorem) every convex polytope is the convex hull of its vertices. It is the unique convex polytope whose vertices belong to
1184:
6993:
6511:
5682:
4582:
3159:
can be used to show that, for large markets, this approximation is accurate, and leads to a "quasi-equilibrium" for the original non-convex market.
413:. This formulation does not immediately generalize to higher dimensions: for a finite set of points in three-dimensional space, a neighborhood of a
6829:
6111:
5122:
4483:
1938:. Reflecting a pocket across its convex hull edge expands the given simple polygon into a polygon with the same perimeter and larger area, and the
6865:
Williams, Jason; Rossignac, Jarek (2005), "Tightening: curvature-limiting morphological simplification", in
Kobbelt, Leif; Shapiro, Vadim (eds.),
2934:
of multivariate polynomials are convex hulls of points derived from the exponents of the terms in the polynomial, and can be used to analyze the
5344:
2824:
to the parameter alpha. The point set itself forms one endpoint of this family of shapes, and its convex hull forms the other endpoint. The
4401:
2584:
is the smallest affine subspace of a
Euclidean space containing a given set, or the union of all affine combinations of points in the set.
2255:
of the hull. In two dimensions, it may suffice more simply to list the points that are vertices, in their cyclic order around the hull.
2591:
is the smallest linear subspace of a vector space containing a given set, or the union of all linear combinations of points in the set.
2018:, there will be times during the Brownian motion where the moving particle touches the boundary of the convex hull at a point of angle
7197:
2916:
Partition of seven points into three subsets with intersecting convex hulls, guaranteed to exist for any seven points in the plane by
1926:
encloses the given polygon and is partitioned by it into regions, one of which is the polygon itself. The other regions, bounded by a
707:
5519:
Kashiwabara, Kenji; Nakamura, Masataka; Okamoto, Yoshio (2005), "The affine representation theorem for abstract convex geometries",
2691:
is the intersection of all relatively convex supersets, where a set within the same polygon is relatively convex if it contains the
6336:
2516:
data structures can be used to keep track of the convex hull of a set of points undergoing insertions and deletions of points, and
2506:, matching the worst-case output complexity of the problem. The convex hull of a simple polygon in the plane can be constructed in
1321:
extends this theory from finite convex combinations of extreme points to infinite combinations (integrals) in more general spaces.
5375:
7097:
5010:
4742:
2103:
or finite set of space curves in general position in three-dimensional space, the parts of the boundary away from the curves are
1314:
5452:(1992), "Hyperconvex hulls of metric spaces", Proceedings of the Symposium on General Topology and Applications (Oxford, 1989),
2239:
of the required convex shape. Output representations that have been considered for convex hulls of point sets include a list of
2119:, the convex hull of two semicircles in perpendicular planes with a common center, and D-forms, the convex shapes obtained from
5716:
2258:
For convex hulls in two or three dimensions, the complexity of the corresponding algorithms is usually estimated in terms of
1934:. Computing the same decomposition recursively for each pocket forms a hierarchical description of a given polygon called its
99:
problems of finding the convex hull of a finite set of points in the plane or other low-dimensional
Euclidean spaces, and its
60:
that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a
7189:
6346:
6255:
6064:
5428:
4908:
6867:
Proceedings of the Tenth ACM Symposium on Solid and
Physical Modeling 2005, Cambridge, Massachusetts, USA, June 13-15, 2005
6264:
Rappoport, Ari (1992), "An efficient adaptive algorithm for constructing the convex differences tree of a simple polygon",
5045:, London Mathematical Society Lecture Note Series, vol. 111, Cambridge: Cambridge University Press, pp. 113–253,
4782:
4707:
72:
subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.
2996:
as well as to
Euclidean spaces. However, in hyperbolic space, it is also possible to consider the convex hulls of sets of
6521:
5489:
3117:
of weights by finding and checking each convex hull vertex, often more efficiently than checking all possible solutions.
3113:, a different type of convex hull is also used, the convex hull of the weight vectors of solutions. One can maximize any
7202:
2120:
460:
It is not obvious that the first definition makes sense: why should there exist a unique minimal convex set containing
4895:, Contemporary Mathematics, vol. 453, Providence, Rhode Island: American Mathematical Society, pp. 231–255,
6488:
6218:
5930:
5734:
5577:
5333:
5259:(1873), "A method of geometrical representation of the thermodynamic properties of substances by means of surfaces",
5194:
5111:
3202:, the study of animal behavior, where it is a classic, though perhaps simplistic, approach in estimating an animal's
6240:
Mathematical
Programming: The State of the Art (XIth International Symposium on Mathematical Programming, Bonn 1982)
5521:
4446:
3235:
of any quantum system — the set of all ways the system can be prepared — is a convex hull whose extreme points are
2938:
behavior of the polynomial and the valuations of its roots. Convex hulls and polynomials also come together in the
2427:
1391:
424:
For objects in three dimensions, the first definition states that the convex hull is the smallest possible convex
7451:
7222:
7139:
1909:
2459:
1833:
6979:
6604:
6177:
6003:
5634:
5212:
4884:
4620:
4349:
2598:
or positive hull of a subset of a vector space is the set of all positive combinations of points in the subset.
1981:
1636:
1278:
The compactness of convex hulls of compact sets, in finite-dimensional
Euclidean spaces, is generalized by the
19:
This article is about the smallest convex shape enclosing a given shape. For boats whose hulls are convex, see
3283:(1873), although the paper was published before the convex hull was so named. In a set of energies of several
3240:
2652:
The circular hull or alpha-hull of a subset of the plane is the intersection of all disks with a given radius
1166:. The points on or above the red curve provide an example of a closed set whose convex hull is open (the open
6945:
6745:
6740:
6294:
5454:
4930:
3216:
3243:
proves that any mixed state can in fact be written as a convex combination of pure states in multiple ways.
149:
for two or three dimensional point sets, and in time matching the worst-case output complexity given by the
7456:
3236:
3110:
580:
is included among the sets being intersected. Thus, it is exactly the unique minimal convex set containing
7242:
3182:
is a measurement of the size of a sailing vessel, defined using the convex hull of a cross-section of the
6940:
6475:, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge: Cambridge University Press,
3156:
3058:. The outer shaded region is the convex hull, and the inner shaded region is the 50% Tukey depth contour.
2780:
2721:
2318:. For higher-dimensional hulls, the number of faces of other dimensions may also come into the analysis.
3296:
The lower convex hull of points in the plane appears, in the form of a Newton polygon, in a letter from
7441:
7207:
6731:(1986), "An optimal algorithm for computing the relative convex hull of a set of points in a polygon",
6446:
3094:
2885:
2848:
Convex hulls have wide applications in many fields. Within mathematics, convex hulls are used to study
2381:
2115:, the convex hull of two circles in perpendicular planes, each passing through the other's center, the
643:, so the set of all convex combinations is contained in the intersection of all convex sets containing
4774:
2751:
1705:
7247:
7237:
6777:
6424:
6035:
5775:
5037:(1987), "Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces", in
3152:
3005:
2725:
2045:
1591:
1279:
856:
and in three-dimensional space it is a tetrahedron. Therefore, every convex combination of points of
393:
and then releasing it, allowing it to contract; when it becomes taut, it encloses the convex hull of
6720:
4840:
Cranston, M.; Hsu, P.; March, P. (1989), "Smoothness of the convex hull of planar Brownian motion",
4796:
4545:
3328:). Other terms, such as "convex envelope", were also used in this time frame. By 1938, according to
1939:
421:
of finding a minimum-energy surface above a given shape can have the convex hull as their solution.
216:
if it contains the line segments connecting each pair of its points. The convex hull of a given set
7446:
7227:
7212:
7054:
5535:
5415:, Topics in Current Chemistry, vol. 345, Springer International Publishing, pp. 139–179,
3098:
3082:
3016:
3009:
2889:
2877:
1310:
1299:
1954:
in the plane, at any fixed time, has probability 1 of having a convex hull whose boundary forms a
1438:
432:, and the definition using convex combinations may be extended from Euclidean spaces to arbitrary
7297:
7274:
7168:
7092:
6935:
6238:(1983), "Polyhedral combinatorics", in Bachem, Achim; Korte, Bernhard; Grötschel, Martin (eds.),
5276:
3190:, the perimeter of the cross-section itself, except for boats and ships that have a convex hull.
2946:
of the derivative of a polynomial all lie within the convex hull of the roots of the polynomial.
2939:
20:
6109:
Oberman, Adam M. (2007), "The convex envelope is the solution of a nonlinear obstacle problem",
5213:"A local nearest-neighbor convex-hull construction of home ranges and utilization distributions"
2387:
2345:
114:
7415:
7376:
7292:
7217:
7144:
7129:
7082:
6715:
6355:
6235:
5530:
4791:
4540:
3148:
3132:
3126:
2989:
concern the existence of partitions of point sets into subsets with intersecting convex hulls.
2893:
2737:
2710:
2699:
2655:
2552:
2232:
2226:
2162:
429:
181:
173:
169:
108:
7361:
7353:
7349:
7345:
7341:
7337:
5987:
5443:
5058:
7436:
7149:
6379:
6198:
5855:
5484:
5291:
5093:
4842:
4531:
4393:
4373:
4337:
3471:, p. 6. The idea of partitioning the hull into two chains comes from an efficient variant of
2986:
2966:
2917:
2729:
2138:
2132:
2021:
1961:
962:
104:
6362:, Princeton Mathematical Series, vol. 28, Princeton, N.J.: Princeton University Press,
2748:, are mathematically related to convex hulls: the Delaunay triangulation of a point set in
2433:
7154:
7020:
6985:
6899:
6858:
6800:
6768:
6700:
6668:
6627:
6576:
6552:
6498:
6408:
6367:
6317:
6228:
6164:
6134:
6082:
6024:
5971:
5940:
5901:
Laurentini, A. (1994), "The visual hull concept for silhouette-based image understanding",
5894:
5853:; Ĺ mulian, V. (1940), "On regularly convex sets in the space conjugate to a Banach space",
5744:
5705:
5653:
5595:
5552:
5512:
5477:
5390:
5367:
5312:
5256:
5229:
5204:
5151:
5050:
5001:
4997:
4990:
4971:
4959:
4918:
4873:
4763:
4730:
4698:
4611:
4562:
4514:
4467:
4422:
4385:
3280:
3114:
2684:
2605:
of a three-dimensional object, with respect to a set of viewpoints, consists of the points
2566:
1955:
819:
5809:
8:
7087:
7077:
7072:
6647:
5038:
5030:
4705:
Chang, J. S.; Yap, C.-K. (1986), "A polynomial solution for the potato-peeling problem",
4150:. See in particular Section 16.9, Non Convexity and Approximate Equilibrium, pp. 209–210.
3136:
3039:
3020:
2935:
2517:
2513:
2423:
2104:
2039:
1787:
1780:
1675:
931:
773:
342:
150:
6086:
5657:
5599:
5383:
Optimizing methods in statistics (Proc. Sympos., Ohio State Univ., Columbus, Ohio, 1971)
5271:
5233:
4347:
Andrew, A. M. (1979), "Another efficient algorithm for convex hulls in two dimensions",
3070:, a method for visualizing the spread of two-dimensional sample points. The contours of
1130:
1083:
623:
must (by the assumption that it is convex) contain all convex combinations of points in
7316:
7034:
6916:
6886:
6846:
6816:
6728:
6707:
6672:
6631:
6556:
6530:
6470:
6396:
6321:
6281:
6098:
5975:
5882:
5828:
5792:
5748:
5669:
5643:
5616:
5561:
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
5394:
5245:
5155:
5139:
5081:
4947:
4888:
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4637:
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4566:
4522:
4502:
4426:
3168:
3144:
3106:
3024:
2982:
2943:
2741:
2628:
2608:
2325:
2301:
2281:
2261:
2192:
2168:
2144:
1883:
1813:
1793:
1758:
1738:
1553:
1533:
1513:
1484:
1464:
1418:
1398:
1369:
1349:
1106:
1063:
1043:
1012:
988:
968:
944:
923:
879:
859:
799:
753:
733:
713:
686:
666:
646:
626:
606:
583:
563:
543:
523:
503:
483:
463:
396:
376:
352:
317:
291:
285:
265:
243:
219:
100:
65:
5319:
4833:
4497:
4459:
3085:
is the convex hull of the risk points of its underlying deterministic decision rules.
2705:
The orthogonal convex hull is a special case of a much more general construction, the
1880:. In particular, in two and three dimensions the number of faces is at most linear in
7134:
6975:
6954:
6791:
6759:
6676:
6595:
6505:
6484:
6459:
6342:
6325:
6251:
6214:
6205:, Algorithms and Computation in Mathematics, vol. 11, Springer, pp. 12–13,
6190:
6060:
6016:
5926:
5796:
5730:
5621:
5573:
5503:
5468:
5434:
5424:
5329:
5304:
5241:
5190:
5174:
5159:
5107:
5085:
4904:
4633:
4362:
3212:
3183:
3063:
2978:
2865:
2861:
2646:
2522:
2240:
2210:
1732:
1080:-dimensional, then every point of the hull belongs to an open convex hull of at most
935:
433:
428:
of the objects. The definition using intersections of convex sets may be extended to
6890:
6635:
6560:
6285:
6102:
5979:
5673:
5358:
5249:
4813:
4641:
4430:
3674:
3662:
3638:
3389:
3387:
3385:
750:-dimensional Euclidean space, every convex combination of finitely many points from
341:
in the Euclidean plane, not all on one line, the boundary of the convex hull is the
7287:
7232:
7123:
7118:
6908:
6878:
6870:
6838:
6786:
6754:
6685:"Fixed points for condensing multifunctions in metric spaces with convex structure"
6656:
6613:
6585:
6540:
6476:
6455:
6437:
6433:
6388:
6303:
6273:
6243:
6206:
6186:
6150:
6120:
6090:
6052:
6012:
5959:
5910:
5872:
5864:
5837:
5805:
5784:
5770:
5756:
5752:
5722:
5691:
5661:
5611:
5603:
5540:
5498:
5463:
5416:
5353:
5323:
5300:
5237:
5182:
5170:
5131:
5099:
5073:
5019:
4967:
4939:
4896:
4851:
4829:
4801:
4770:
4751:
4737:
4716:
4686:
4652:
4629:
4591:
4577:
4570:
4550:
4526:
4492:
4474:
4455:
4410:
4358:
3794:
3353:
3321:
3305:
3151:
can be used to prove the existence of an equilibrium. When actual economic data is
3102:
2993:
2881:
2836:
of a polygon is the largest convex polygon contained inside it. It can be found in
2706:
2298:, the number of points on the convex hull, which may be significantly smaller than
2248:
2244:
2214:
2123:
for a surface formed by gluing together two planar convex sets of equal perimeter.
1776:
1335:
1272:
1167:
663:. Conversely, the set of all convex combinations is itself a convex set containing
441:
418:
95:
can be represented by applying this closure operator to finite sets of points. The
88:
6684:
6599:
6544:
6125:
5696:
1914:
7307:
7278:
7252:
7173:
7158:
7067:
7039:
7016:
6854:
6796:
6764:
6696:
6664:
6623:
6572:
6548:
6494:
6415:
6404:
6363:
6332:
6313:
6247:
6224:
6160:
6130:
6056:
6020:
5967:
5936:
5890:
5801:
5740:
5721:, Lecture Notes in Computer Science, vol. 606, Heidelberg: Springer-Verlag,
5701:
5548:
5544:
5508:
5473:
5386:
5363:
5342:
Gustin, William (1947), "On the interior of the convex hull of a Euclidean set",
5308:
5200:
5147:
5046:
4986:
4955:
4914:
4869:
4759:
4726:
4694:
4672:
4607:
4558:
4510:
4463:
4418:
4381:
3382:
3317:
3313:
3301:
3228:
3078:
2970:
2954:
2950:
2931:
2857:
2837:
2745:
2649:
as the largest shape that could have the same outlines from the given viewpoints.
2186:
1951:
1927:
1699:
1628:
1603:
1268:
1163:
500:? However, the second definition, the intersection of all convex sets containing
425:
209:
185:
161:
61:
6971:
6958:
6733:
Proceedings of EURASIP, Signal Processing III: Theories and Applications, Part 2
3172:
2897:
1257:{\displaystyle \left\{(x,y)\mathop {\bigg |} y\geq {\frac {1}{1+x^{2}}}\right\}}
7394:
7302:
7163:
7062:
6642:
6519:
Seaton, Katherine A. (2017), "Sphericons and D-forms: a crocheted connection",
6466:
5607:
5449:
5166:
5005:
4900:
4660:
4441:
3698:
3276:
3013:
2923:
2688:
2236:
1923:
1669:
1318:
1124:
157:
6277:
6210:
6094:
6001:(1979), "A linear algorithm for finding the convex hull of a simple polygon",
5877:
5680:
Kiselman, Christer O. (2002), "A semigroup of operators in convexity theory",
5665:
5186:
4690:
3004:
in Euclidean space, and their metric properties play an important role in the
1123:. The sets of vertices of a square, regular octahedron, or higher-dimensional
7430:
7178:
6897:
Worton, Bruce J. (1995), "A convex hull-based estimator of home-range size",
6480:
5282:
5034:
5023:
4856:
4755:
4656:
3284:
3001:
2958:
2825:
2108:
1587:
1305:
1287:
414:
177:
84:
6874:
6660:
6618:
5120:
Gardner, L. Terrell (1984), "An elementary proof of the Russo-Dye theorem",
5043:
Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984)
4478:
3912:
3910:
3239:
known as pure states and whose interior points are called mixed states. The
3012:. Hyperbolic convex hulls have also been used as part of the calculation of
1282:, according to which the closed convex hull of a weakly compact subset of a
7399:
6031:
5947:
5842:
5819:
5712:
5625:
5438:
4880:
4648:
4554:
3413:
3297:
3251:
2833:
2821:
2714:
2595:
2252:
1574:
When applied to a finite set of points, this is the closure operator of an
1283:
437:
189:
5726:
5413:
Prediction and Calculation of Crystal Structures: Methods and Applications
5103:
4820:
Chen, Qinyu; Wang, Guozhao (March 2003), "A class of BĂ©zier-like curves",
156:
As well as for finite point sets, convex hulls have also been studied for
7389:
7384:
6645:(1914), "Bedingt konvergente Reihen und konvexe Systeme. (Fortsetzung)",
6175:(1984), "On the definition and computation of rectilinear convex hulls",
6172:
5648:
5420:
5328:, Graduate Texts in Mathematics, vol. 221 (2nd ed.), Springer,
5286:
4925:
3907:
3472:
3329:
3232:
3179:
3071:
2997:
2869:
2816:
2602:
2588:
2581:
2546:
2507:
2319:
2100:
1575:
1506:
1034:
370:
338:
165:
92:
80:
69:
6820:
6743:(1993), "Convex hulls and isometries of cusped hyperbolic 3-manifolds",
5181:, Mathematics: Theory & Applications, Birkhäuser, pp. 193–213,
3934:
3409:
3034:
for the application of convex hulls to this subject, and the section on
1313:, every compact convex set in a Euclidean space (or more generally in a
7312:
7282:
7044:
6920:
6850:
6589:
6567:
Sedykh, V. D. (1981), "Structure of the convex hull of a space curve",
6419:
6400:
6374:
6308:
6292:
Reay, John R. (1979), "Several generalizations of Tverberg's theorem",
5998:
5963:
5886:
5850:
5815:
5788:
5143:
4951:
4865:
4805:
4721:
4603:
4506:
4437:
4369:
3304:
in 1676. The term "convex hull" itself appears as early as the work of
3203:
3187:
3140:
2974:
2962:
2927:
2901:
2896:
to non-convex markets. In geometric modeling, the convex hull property
2853:
2849:
2560:
1030:
1006:
213:
57:
33:
6155:
5914:
3610:
6963:
6882:
5220:
5077:
4647:
4414:
3816:
3800:
3680:
3668:
3644:
3468:
3393:
3256:
2116:
683:, so it also contains the intersection of all convex sets containing
346:
96:
6912:
6842:
6392:
6377:(1961), "Holomorphically convex sets in several complex variables",
5868:
5135:
4943:
4595:
2912:
7321:
6535:
4879:
3704:
3199:
3198:
The convex hull is commonly known as the minimum convex polygon in
2692:
2530:
1175:
853:
76:
41:
4580:(1935), "Integration of functions with values in a Banach space",
3854:
2087:
452:
440:; convex hulls may also be generalized in a more abstract way, to
7102:
6710:(1983), "Solving geometric problems with the rotating calipers",
5559:
Katoh, Naoki (1992), "Bicriteria network optimization problems",
5272:
The Scientific Papers of J. Willard Gibbs, Vol. I: Thermodynamics
3207:
3067:
3055:
2873:
849:
703:, and therefore the second and third definitions are equivalent.
311:
28:
4678:
International Journal of Computational Geometry and Applications
1334:
The convex-hull operator has the characteristic properties of a
83:
are compact. Every compact convex set is the convex hull of its
5823:
3260:
2876:
visualization of two-dimensional data, and define risk sets of
1918:
Convex hull ( in blue and yellow) of a simple polygon (in blue)
1594:
bounding the distance of a Minkowski sum from its convex hull.
5903:
IEEE Transactions on Pattern Analysis and Machine Intelligence
5165:
4093:
4016:
1619:
1586:
The operations of constructing the convex hull and taking the
1158:
6047:
Nicola, Piercarlo (2000), "General Competitive Equilibrium",
4996:
3916:
3782:
3356:. However, this term is also frequently used to refer to the
2892:. In economics, convex hulls can be used to apply methods of
2526:
2112:
2092:
200:
7015:
6170:
5261:
Transactions of the Connecticut Academy of Arts and Sciences
5179:
Discriminants, Resultants, and Multidimensional Determinants
3940:
3710:
2042:
of this set of exceptional times is (with high probability)
520:, is well-defined. It is a subset of every other convex set
4618:
Brown, K. Q. (1979), "Voronoi diagrams from convex hulls",
3734:
3372:
3370:
3271:
is expected to be unstable as it lies above the lower hull.
2380:. For points in two and three dimensions, more complicated
5952:
International Journal of Computer and Information Sciences
5950:(1983), "On finding the convex hull of a simple polygon",
5632:
Kirkpatrick, K. A. (2006), "The Schrödinger–HJW theorem",
5518:
3770:
3616:
3066:, the convex hull provides one of the key components of a
3050:
1942:
states that this expansion process eventually terminates.
5059:"Convex polytopes, algebraic geometry, and combinatorics"
4775:"An optimal convex hull algorithm in any fixed dimension"
4339:
Convex Sets and Their Applications. Summer Lectures 1959.
4069:
3885:
3883:
3881:
3758:
3635:, Theorem 1.1.2 (pages 2–3) and Chapter 3.
3494:
1930:
of the polygon and a single convex hull edge, are called
1153:
5567:
5008:(1983), "On the shape of a set of points in the plane",
4240:
4216:
4183:
3367:
204:
Convex hull of a bounded planar set: rubber band analogy
5289:(1983), "Finding the convex hull of a simple polygon",
4228:
4204:
896:, and the third and fourth definitions are equivalent.
600:. Therefore, the first two definitions are equivalent.
4520:
3895:
3878:
3860:
3650:
2884:
of solutions to combinatorial problems are central to
6827:
Whitley, Robert (1986), "The KreÄn-Ĺ mulian theorem",
6775:
Westermann, L. R. J. (1976), "On the hull operator",
6072:
6049:
Mainstream Mathematical Economics in the 20th Century
5376:"Mathematical models for statistical decision theory"
4675:(2012), "Three problems about dynamic convex hulls",
4187:
3922:
3866:
3686:
2783:
2754:
2658:
2631:
2611:
2462:
2436:
2390:
2348:
2328:
2304:
2284:
2264:
2195:
2171:
2147:
2048:
2024:
1984:
1964:
1886:
1836:
1816:
1796:
1761:
1741:
1708:
1678:
1639:
1556:
1536:
1516:
1487:
1467:
1441:
1421:
1401:
1372:
1352:
1324:
1187:
1133:
1109:
1086:
1066:
1046:
1015:
991:
971:
947:
882:
862:
822:
802:
776:
756:
736:
716:
689:
669:
649:
629:
609:
586:
566:
546:
526:
506:
486:
466:
399:
379:
355:
320:
294:
268:
246:
222:
117:
6952:
6444:
Sakuma, Itsuo (1977), "Closedness of convex hulls",
6141:
Okon, T. (2000), "Choquet theory in metric spaces",
4473:
4153:
4021:
3994:
3982:
3946:
3598:
3520:
3206:
based on points where the animal has been observed.
3155:, it can be made convex by taking convex hulls. The
2777:
can be viewed as the projection of a convex hull in
6807:White, F. Puryer (April 1923), "Pure mathematics",
4665:
Computational Geometry: Algorithms and Applications
4045:
4033:
3558:
3101:, central objects of study are the convex hulls of
32:The convex hull of the red set is the blue and red
6414:
5177:(1994), "6. Newton Polytopes and Chow Polytopes",
4192:
4105:
4099:
3746:
3722:
3419:
3352:refers to the fact that the convex hull defines a
2807:
2769:
2728:, obtained as an intersection of sublevel sets of
2672:
2637:
2617:
2498:
2448:
2414:
2372:
2334:
2310:
2290:
2270:
2201:
2177:
2153:
2071:
2030:
2010:
1970:
1892:
1872:
1822:
1802:
1767:
1747:
1723:
1690:
1660:
1562:
1542:
1522:
1493:
1473:
1453:
1427:
1407:
1378:
1358:
1256:
1142:
1115:
1095:
1072:
1052:
1021:
997:
977:
953:
888:
868:
840:
808:
788:
762:
742:
722:
695:
675:
655:
635:
615:
592:
572:
552:
532:
512:
492:
472:
405:
385:
361:
326:
300:
274:
252:
228:
141:
6864:
5683:Transactions of the American Mathematical Society
5057:Escobar, Laura; Kaveh, Kiumars (September 2020),
4891:(2008), "All polygons flip finitely ... right?",
4583:Transactions of the American Mathematical Society
4436:
4312:
4252:
4165:
4129:
4081:
3970:
3958:
3788:
3482:
3405:
3360:, with which it should not be confused — see e.g
1210:
7428:
6830:Proceedings of the American Mathematical Society
6112:Proceedings of the American Mathematical Society
5211:Getz, Wayne M.; Wilmers, Christopher C. (2004),
5123:Proceedings of the American Mathematical Society
4839:
4740:(1985), "On the convex layers of a planar set",
4484:Proceedings of the American Mathematical Society
4444:(1997), "How good are convex hull algorithms?",
4057:
3842:
3716:
3586:
3431:
6648:Journal fĂĽr die Reine und Angewandte Mathematik
4965:
4126:; see especially remarks following Theorem 2.9.
3740:
2384:are known that compute the convex hull in time
262:The intersection of all convex sets containing
4893:Surveys on Discrete and Computational Geometry
3023:, and applied to determine the equivalence of
876:belongs to a simplex whose vertices belong to
87:. The convex hull operator is an example of a
7001:
6510:: CS1 maint: DOI inactive as of March 2024 (
6143:Zeitschrift fĂĽr Analysis und ihre Anwendungen
5487:(1976), "Normality and the numerical range",
5345:Bulletin of the American Mathematical Society
5029:
4075:
3917:Edelsbrunner, Kirkpatrick & Seidel (1983)
3448:
3446:
3410:answer to "the perimeter of a non-convex set"
3088:
2840:, but the exponent of the algorithm is high.
2625:such that every ray from a viewpoint through
2251:of facets and their adjacencies, or the full
2161:on a real vector space is the function whose
1731:. Each extreme point of the hull is called a
447:
6422:(1999), "The bagplot: A bivariate boxplot",
6331:
5996:
5849:
5585:
5066:Notices of the American Mathematical Society
5056:
4481:(1982), "Quantitative Helly-type theorems",
4402:Notices of the American Mathematical Society
4293:
4246:
4222:
3941:Ottmann, Soisalon-Soininen & Wood (1984)
3828:
3628:
2488:
2474:
2456:, the time for computing the convex hull is
2165:is the lower convex hull of the epigraph of
2095:, the convex hull of two circles in 3d space
1862:
1848:
1790:, the number of faces of the convex hull of
1346:, meaning that the convex hull of every set
1267:(the set of points that lie on or above the
6354:
6234:
5814:
5631:
5210:
4529:(1999), "Data structures for mobile data",
4234:
4210:
4123:
3776:
3764:
3576:
3500:
3452:
3376:
2724:is a generalization of similar concepts to
240:The (unique) minimal convex set containing
7008:
6994:
6774:
5900:
5824:"On extreme points of regular convex sets"
5098:, Cambridge University Press, p. 55,
4017:Gel'fand, Kapranov & Zelevinsky (1994)
3901:
3889:
3617:Kashiwabara, Nakamura & Okamoto (2005)
3443:
3109:can be used to find optimal solutions. In
2709:, which can be thought of as the smallest
2499:{\displaystyle O(n^{\lfloor d/2\rfloor })}
1873:{\displaystyle O(n^{\lfloor d/2\rfloor })}
16:Smallest convex set containing a given set
6809:Science Progress in the Twentieth Century
6790:
6758:
6727:
6719:
6706:
6617:
6534:
6472:Convex Bodies: The Brunn–Minkowski Theory
6465:
6307:
6263:
6154:
6124:
5876:
5841:
5695:
5647:
5615:
5534:
5502:
5467:
5357:
5281:
4855:
4795:
4720:
4544:
4496:
4378:Algebraic Numbers and Algebraic Functions
3928:
3872:
3832:
3692:
3632:
3178:In the geometry of boat and ship design,
2786:
2757:
2011:{\displaystyle \pi /2<\theta <\pi }
1711:
1661:{\displaystyle S\subset \mathbb {R} ^{d}}
1648:
908:
6641:
6338:Quantum Computing: A Gentle Introduction
6196:
5773:(December 1922), "Über konvexe Körper",
5679:
5448:
5318:
4819:
4769:
4736:
4704:
4576:
4394:"The mathematics of Grace Murray Hopper"
4184:Kernohan, Gitzen & Millspaugh (2001)
4159:
4027:
4000:
3988:
3952:
3812:
3656:
3604:
3531:
3512:
3309:
3250:
3049:
2911:
2086:
1913:
1618:
1530:, the convex hull of the convex hull of
1157:
913:
899:
770:is also a convex combination of at most
451:
199:
27:
7098:Locally convex topological vector space
6826:
6712:Proceedings of IEEE MELECON '83, Athens
6108:
5483:
5410:
5119:
5011:IEEE Transactions on Information Theory
4743:IEEE Transactions on Information Theory
4270:
4051:
4039:
3564:
3425:
3038:for their application to the theory of
2864:involve convex hulls. They are used in
1315:locally convex topological vector space
7429:
6896:
6682:
6594:
6569:Trudy Seminara imeni I. G. Petrovskogo
6566:
6518:
6443:
6046:
6030:
5373:
5341:
4346:
4285:
4198:
4147:
4111:
3861:Basch, Guibas & Hershberger (1999)
3752:
3728:
3631:, Theorem 3, pages 562–563;
3552:
3539:
3516:
3488:
3476:
1633:The convex hull of a finite point set
1154:Preservation of topological properties
816:. The set of convex combinations of a
6989:
6953:
6806:
6739:
6600:"Remarks on piecewise-linear algebra"
6373:
5985:
5769:
5711:
5570:Radio Tracking and Animal Populations
5558:
5255:
5091:
4924:
4783:Discrete & Computational Geometry
4708:Discrete & Computational Geometry
4617:
4380:, Gordon and Breach, pp. 37–43,
4368:
4318:
4306:
4274:
4258:
4188:Nilsen, Pedersen & Linnell (2008)
4171:
4135:
4087:
4012:
3976:
3964:
3437:
3325:
3162:
2536:
1597:
1174:Topologically, the convex hull of an
1009:itself (as happens, for instance, if
6291:
6171:Ottmann, T.; Soisalon-Soininen, E.;
6140:
5275:, Longmans, Green, & Co., 1906,
4671:
4391:
4289:
4063:
3848:
3592:
3521:Bárány, Katchalski & Pach (1982)
3215:method by combining convex hulls of
3139:, agents are assumed to have convex
3031:
2213:) and, in this form, is dual to the
1614:
1286:(a subset that is compact under the
373:so that it surrounds the entire set
64:, or equivalently as the set of all
21:Hull (watercraft) § Hull shapes
6522:Journal of Mathematics and the Arts
5989:Encyclopaedia of Ships and Shipping
5946:
5920:
5490:Linear Algebra and Its Applications
4335:
3836:
3580:
3535:
3456:
3361:
3186:of the vessel. It differs from the
2808:{\displaystyle \mathbb {R} ^{n+1}.}
1395:, meaning that, for every two sets
1329:
13:
6689:KĹŤdai Mathematical Seminar Reports
5923:Convex Sets and their Applications
4100:Rousseeuw, Ruts & Tukey (1999)
3222:
2278:, the number of input points, and
1945:
1903:
1623:Convex hull of points in the plane
1550:is the same as the convex hull of
1481:is a subset of the convex hull of
1325:Geometric and algebraic properties
961:is the intersection of all closed
14:
7468:
6928:
6735:, North-Holland, pp. 853–856
4979:Journal for Geometry and Graphics
4498:10.1090/S0002-9939-1982-0663877-X
3789:Avis, Bremner & Seidel (1997)
3316:appears earlier, for instance in
3312:), and the corresponding term in
3246:
3171:, one of the key properties of a
3035:
2713:containing the points of a given
2687:of a subset of a two-dimensional
1956:continuously differentiable curve
1293:
1040:If the open convex hull of a set
456:3D convex hull of 120 point cloud
5242:10.1111/j.0906-7590.2004.03835.x
3717:Cranston, Hsu & March (1989)
2770:{\displaystyle \mathbb {R} ^{n}}
2559:
2545:
1830:-dimensional Euclidean space is
1724:{\displaystyle \mathbb {R} ^{d}}
1609:
1581:
7203:Ekeland's variational principle
6341:, MIT Press, pp. 215–216,
6199:"1.2.1 The Gauss–Lucas theorem"
5359:10.1090/S0002-9904-1947-08787-5
4822:Computer Aided Geometric Design
4299:
4279:
4264:
4177:
4141:
4117:
4006:
3822:
3806:
3622:
3570:
3545:
3525:
3506:
3406:Williams & Rossignac (2005)
3237:positive-semidefinite operators
2843:
2322:can compute the convex hull of
2121:Alexandrov's uniqueness theorem
2082:
2072:{\displaystyle 1-\pi /2\theta }
1910:Convex hull of a simple polygon
1127:provide examples where exactly
369:. One may imagine stretching a
68:of points in the subset. For a
6980:Wolfram Demonstrations Project
6605:Pacific Journal of Mathematics
6438:10.1080/00031305.1999.10474494
6242:, Springer, pp. 312–345,
6051:, Springer, pp. 197–215,
6004:Information Processing Letters
5635:Foundations of Physics Letters
5095:Phase Transitions in Materials
4972:"The development of the oloid"
4621:Information Processing Letters
4350:Information Processing Letters
3462:
3399:
3342:
2969:describes the convex hulls of
2907:
2695:between any two of its points.
2493:
2466:
2409:
2394:
2367:
2352:
2220:
1867:
1840:
1205:
1193:
835:
823:
195:
136:
121:
107:, are fundamental problems of
79:are open, and convex hulls of
1:
6746:Topology and Its Applications
6582:Journal of Soviet Mathematics
6545:10.1080/17513472.2017.1318512
6335:; Polak, Wolfgang H. (2011),
6295:Israel Journal of Mathematics
6126:10.1090/S0002-9939-07-08887-9
5697:10.1090/S0002-9947-02-02915-X
5455:Topology and Its Applications
4931:American Mathematical Monthly
4834:10.1016/s0167-8396(03)00003-7
4460:10.1016/S0925-7721(96)00023-5
4328:
3741:Dirnböck & Stachel (1997)
3045:
111:. They can be solved in time
6792:10.1016/1385-7258(76)90065-2
6760:10.1016/0166-8641(93)90032-9
6460:10.1016/0022-0531(77)90095-3
6248:10.1007/978-3-642-68874-4_13
6197:Prasolov, Victor V. (2004),
6191:10.1016/0020-0255(84)90025-2
6057:10.1007/978-3-662-04238-0_16
6017:10.1016/0020-0190(79)90069-3
5545:10.1016/j.comgeo.2004.05.001
5504:10.1016/0024-3795(76)90080-x
5469:10.1016/0166-8641(92)90092-E
5305:10.1016/0196-6774(83)90013-5
4634:10.1016/0020-0190(79)90074-7
4363:10.1016/0020-0190(79)90072-3
3137:general economic equilibrium
3120:
3111:multi-objective optimization
2868:as the outermost contour of
2428:Kirkpatrick–Seidel algorithm
2342:points in the plane in time
2126:
1454:{\displaystyle X\subseteq Y}
926:of the convex hull, and the
7:
7223:Hermite–Hadamard inequality
6941:Encyclopedia of Mathematics
6036:"Letter to Henry Oldenburg"
5092:Fultz, Brent (April 2020),
4342:, Argon national laboratory
4076:Epstein & Marden (1987)
3193:
2722:holomorphically convex hull
2382:output-sensitive algorithms
2185:. It is the unique maximal
603:Each convex set containing
56:of a shape is the smallest
10:
7473:
6447:Journal of Economic Theory
5986:Mason, Herbert B. (1908),
5608:10.1038/s41598-019-56497-6
4294:Escobar & Kaveh (2020)
4223:Rieffel & Polak (2011)
3829:McCallum & Avis (1979)
3629:Krein & Ĺ mulian (1940)
3291:
3124:
3095:combinatorial optimization
3089:Combinatorial optimization
2961:is the convex hull of its
2886:combinatorial optimization
2860:, and several theorems in
2726:complex analytic manifolds
2415:{\displaystyle O(n\log h)}
2373:{\displaystyle O(n\log n)}
2224:
2130:
1907:
1626:
1297:
941:The closed convex hull of
448:Equivalence of definitions
142:{\displaystyle O(n\log n)}
18:
7408:
7375:
7330:
7261:
7187:
7111:
7053:
7027:
6869:, ACM, pp. 107–112,
6778:Indagationes Mathematicae
6683:Talman, Louis A. (1977),
6584:33 (4): 1140–1153, 1986,
6425:The American Statistician
6278:10.1111/1467-8659.1140235
6211:10.1007/978-3-642-03980-5
6095:10.1007/s11284-007-0421-9
5925:, John Wiley & Sons,
5776:Mathematische Zeitschrift
5666:10.1007/s10702-006-1852-1
5187:10.1007/978-0-8176-4771-1
4691:10.1142/S0218195912600096
4211:Getz & Wilmers (2004)
3577:Krein & Milman (1940)
3551:This example is given by
3408:. See also Douglas Zare,
3006:geometrization conjecture
2942:, according to which the
2878:randomized decision rules
2673:{\displaystyle 1/\alpha }
2525:method for computing the
2099:For the convex hull of a
1958:. However, for any angle
1755:and that encloses all of
1510:, meaning that for every
7409:Applications and related
7213:Fenchel-Young inequality
6481:10.1017/CBO9780511526282
5374:Harris, Bernard (1971),
5024:10.1109/TIT.1983.1056714
4928:(1938), "On convexity",
4756:10.1109/TIT.1985.1057060
4667:(3rd ed.), Springer
3335:
3099:polyhedral combinatorics
3083:randomized decision rule
3030:See also the section on
3010:low-dimensional topology
2890:polyhedral combinatorics
2680:that contain the subset.
1775:. For sets of points in
985:. If the convex hull of
934:(or in some sources the
103:problem of intersecting
7169:Legendre transformation
7093:Legendre transformation
6875:10.1145/1060244.1060257
6661:10.1515/crll.1914.144.1
6619:10.2140/pjm.1982.98.183
6483:(inactive 2024-03-18),
6356:Rockafellar, R. Tyrrell
6266:Computer Graphics Forum
5921:Lay, Steven R. (1982),
4374:"2.5. Newton's Polygon"
3833:Graham & Yao (1983)
3241:Schrödinger–HJW theorem
3157:Shapley–Folkman theorem
3147:. These assumptions of
3115:quasiconvex combination
2740:of a point set and its
2732:containing a given set.
2111:. Examples include the
2031:{\displaystyle \theta }
1971:{\displaystyle \theta }
1950:The curve generated by
1936:convex differences tree
1779:, the convex hull is a
1592:Shapley–Folkman theorem
852:; in the plane it is a
7452:Computational geometry
7416:Convexity in economics
7350:(lower) ideally convex
7208:Fenchel–Moreau theorem
7198:Carathéodory's theorem
6042:, University of Oxford
5843:10.4064/sm-9-1-133-138
5522:Computational Geometry
4901:10.1090/conm/453/08801
4889:Toussaint, Godfried T.
4857:10.1214/aop/1176991500
4555:10.1006/jagm.1998.0988
4447:Computational Geometry
4160:Chen & Wang (2003)
4001:Chang & Yap (1986)
3272:
3149:convexity in economics
3127:Convexity in economics
3059:
2920:
2894:convexity in economics
2809:
2771:
2738:Delaunay triangulation
2711:injective metric space
2700:orthogonal convex hull
2674:
2639:
2619:
2553:Orthogonal convex hull
2500:
2450:
2449:{\displaystyle d>3}
2416:
2374:
2336:
2312:
2292:
2272:
2233:computational geometry
2227:Convex hull algorithms
2203:
2179:
2155:
2096:
2073:
2032:
2012:
1972:
1919:
1894:
1874:
1824:
1804:
1769:
1749:
1725:
1698:, or more generally a
1692:
1662:
1624:
1564:
1544:
1524:
1495:
1475:
1455:
1429:
1409:
1380:
1360:
1258:
1171:
1144:
1117:
1097:
1074:
1054:
1023:
999:
979:
955:
938:) of the convex hull.
909:Topological properties
890:
870:
848:-tuple of points is a
842:
810:
790:
764:
744:
724:
708:Carathéodory's theorem
706:In fact, according to
697:
677:
657:
637:
617:
594:
574:
554:
534:
514:
494:
474:
457:
430:non-Euclidean geometry
407:
387:
363:
328:
302:
276:
254:
230:
205:
182:Delaunay triangulation
174:orthogonal convex hull
170:epigraphs of functions
153:in higher dimensions.
143:
109:computational geometry
37:
7338:Convex series related
7238:Shapley–Folkman lemma
6380:Annals of Mathematics
6302:(3): 238–244 (1980),
5856:Annals of Mathematics
5800:; see also review by
5727:10.1007/3-540-55611-7
5292:Journal of Algorithms
5228:(4), Wiley: 489–505,
5104:10.1017/9781108641449
5002:Kirkpatrick, David G.
4998:Edelsbrunner, Herbert
4843:Annals of Probability
4532:Journal of Algorithms
3817:de Berg et al. (2008)
3801:de Berg et al. (2008)
3705:Demaine et al. (2008)
3681:de Berg et al. (2008)
3669:de Berg et al. (2008)
3645:de Berg et al. (2008)
3469:de Berg et al. (2008)
3394:de Berg et al. (2008)
3254:
3053:
2915:
2810:
2772:
2730:holomorphic functions
2675:
2640:
2620:
2501:
2451:
2417:
2375:
2337:
2313:
2293:
2273:
2204:
2180:
2156:
2139:lower convex envelope
2133:Lower convex envelope
2090:
2074:
2033:
2013:
1973:
1922:The convex hull of a
1917:
1895:
1875:
1825:
1805:
1770:
1750:
1726:
1693:
1663:
1622:
1565:
1545:
1525:
1496:
1476:
1461:, the convex hull of
1456:
1430:
1410:
1381:
1361:
1290:) is weakly compact.
1280:Krein–Smulian theorem
1259:
1161:
1145:
1118:
1098:
1075:
1055:
1024:
1000:
980:
956:
914:Closed and open hulls
900:Upper and lower hulls
891:
871:
843:
841:{\displaystyle (d+1)}
811:
791:
765:
745:
725:
698:
678:
658:
638:
618:
595:
575:
555:
535:
515:
495:
475:
455:
408:
388:
364:
329:
303:
277:
255:
231:
208:A set of points in a
203:
144:
31:
7228:Krein–Milman theorem
7021:variational analysis
6178:Information Sciences
6034:(October 24, 1676),
5421:10.1007/128_2013_486
5385:, pp. 369–389,
4477:; Katchalski, Meir;
4392:Auel, Asher (2019),
3306:Garrett Birkhoff
3281:Josiah Willard Gibbs
3081:, the risk set of a
3040:developable surfaces
3021:hyperbolic manifolds
2781:
2752:
2685:relative convex hull
2656:
2629:
2609:
2567:Relative convex hull
2460:
2434:
2388:
2346:
2326:
2302:
2282:
2262:
2193:
2169:
2145:
2046:
2022:
1982:
1962:
1884:
1834:
1814:
1794:
1759:
1739:
1706:
1676:
1637:
1554:
1534:
1514:
1485:
1465:
1439:
1419:
1399:
1370:
1350:
1311:Krein–Milman theorem
1300:Krein–Milman theorem
1275:as its convex hull.
1185:
1131:
1107:
1084:
1064:
1044:
1033:or more generally a
1013:
989:
969:
945:
880:
860:
820:
800:
774:
754:
734:
714:
687:
667:
647:
627:
607:
584:
564:
544:
524:
504:
484:
464:
397:
377:
353:
318:
292:
266:
244:
220:
115:
7457:Geometry processing
7218:Jensen's inequality
7088:Lagrange multiplier
7078:Convex optimization
7073:Convex metric space
6729:Toussaint, Godfried
6708:Toussaint, Godfried
6416:Rousseeuw, Peter J.
6333:Rieffel, Eleanor G.
6087:2008EcoR...23..635N
6075:Ecological Research
5658:2006FoPhL..19...95K
5600:2019NatSR...920253K
5485:Johnson, Charles R.
5234:2004Ecogr..27..489G
4883:; Gassend, Blaise;
4523:Guibas, Leonidas J.
2940:Gauss–Lucas theorem
2518:kinetic convex hull
2514:Dynamic convex hull
2241:linear inequalities
2137:The convex hull or
2040:Hausdorff dimension
1788:upper bound theorem
1781:simplicial polytope
1691:{\displaystyle d=2}
1150:points are needed.
789:{\displaystyle d+1}
343:simple closed curve
286:convex combinations
151:upper bound theorem
66:convex combinations
7346:(cs, bcs)-complete
7317:Algebraic interior
7035:Convex combination
6955:Weisstein, Eric W.
6596:Sontag, Eduardo D.
6590:10.1007/BF01086114
6309:10.1007/BF02760885
6236:Pulleyblank, W. R.
6040:The Newton Project
5997:McCallum, Duncan;
5964:10.1007/BF00993195
5878:10338.dmlcz/100106
5829:Studia Mathematica
5789:10.1007/bf01215899
5588:Scientific Reports
5572:, Academic Press,
4806:10.1007/BF02573985
4722:10.1007/BF02187692
4440:; Bremner, David;
4235:Kirkpatrick (2006)
4124:Pulleyblank (1983)
3777:Rockafellar (1970)
3765:Rockafellar (1970)
3501:Rockafellar (1970)
3453:Rockafellar (1970)
3377:Rockafellar (1970)
3358:closed convex hull
3279:was identified by
3273:
3169:geometric modeling
3163:Geometric modeling
3145:convex preferences
3133:Arrow–Debreu model
3107:linear programming
3060:
2987:Tverberg's theorem
2921:
2918:Tverberg's theorem
2880:. Convex hulls of
2872:, are part of the
2805:
2767:
2670:
2635:
2615:
2537:Related structures
2496:
2446:
2412:
2370:
2332:
2308:
2288:
2268:
2199:
2175:
2151:
2097:
2069:
2028:
2008:
1968:
1940:Erdős–Nagy theorem
1920:
1890:
1870:
1820:
1800:
1765:
1745:
1721:
1688:
1658:
1625:
1598:Projective duality
1560:
1540:
1520:
1491:
1471:
1451:
1425:
1405:
1376:
1356:
1254:
1172:
1143:{\displaystyle 2d}
1140:
1113:
1096:{\displaystyle 2d}
1093:
1070:
1050:
1019:
995:
975:
951:
920:closed convex hull
886:
866:
838:
806:
786:
760:
740:
720:
693:
673:
653:
633:
613:
590:
570:
550:
530:
510:
490:
470:
458:
434:real vector spaces
403:
383:
359:
324:
298:
272:
250:
236:may be defined as
226:
206:
139:
38:
7442:Closure operators
7424:
7423:
6976:Eric W. Weisstein
6741:Weeks, Jeffrey R.
6383:, Second Series,
6348:978-0-262-01506-6
6257:978-3-642-68876-8
6066:978-3-642-08638-0
5915:10.1109/34.273735
5859:, Second Series,
5430:978-3-319-05773-6
5283:Graham, Ronald L.
5257:Gibbs, Willard J.
5175:Zelevinsky, A. V.
5039:Epstein, D. B. A.
5031:Epstein, D. B. A.
4968:Stachel, Hellmuth
4910:978-0-8218-4239-3
4771:Chazelle, Bernard
4738:Chazelle, Bernard
4578:Birkhoff, Garrett
4527:Hershberger, John
4247:Kim et al. (2019)
3902:Laurentini (1994)
3890:Westermann (1976)
3275:A convex hull in
3213:local convex hull
3103:indicator vectors
3064:robust statistics
2994:hyperbolic spaces
2979:discrete geometry
2967:Russo–Dye theorem
2951:spectral analysis
2882:indicator vectors
2866:robust statistics
2862:discrete geometry
2647:3D reconstruction
2638:{\displaystyle p}
2618:{\displaystyle p}
2523:rotating calipers
2430:. For dimensions
2335:{\displaystyle n}
2311:{\displaystyle n}
2291:{\displaystyle h}
2271:{\displaystyle n}
2211:pointwise minimum
2202:{\displaystyle f}
2178:{\displaystyle f}
2154:{\displaystyle f}
1893:{\displaystyle n}
1823:{\displaystyle d}
1803:{\displaystyle n}
1786:According to the
1768:{\displaystyle S}
1748:{\displaystyle S}
1615:Finite point sets
1563:{\displaystyle X}
1543:{\displaystyle X}
1523:{\displaystyle X}
1494:{\displaystyle Y}
1474:{\displaystyle X}
1428:{\displaystyle Y}
1408:{\displaystyle X}
1379:{\displaystyle X}
1366:is a superset of
1359:{\displaystyle X}
1247:
1116:{\displaystyle X}
1073:{\displaystyle d}
1053:{\displaystyle X}
1022:{\displaystyle X}
998:{\displaystyle X}
978:{\displaystyle X}
954:{\displaystyle X}
936:relative interior
889:{\displaystyle X}
869:{\displaystyle X}
809:{\displaystyle X}
763:{\displaystyle X}
743:{\displaystyle d}
730:is a subset of a
723:{\displaystyle X}
696:{\displaystyle X}
676:{\displaystyle X}
656:{\displaystyle X}
636:{\displaystyle X}
616:{\displaystyle X}
593:{\displaystyle X}
573:{\displaystyle Y}
553:{\displaystyle X}
533:{\displaystyle Y}
513:{\displaystyle X}
493:{\displaystyle X}
473:{\displaystyle X}
442:oriented matroids
406:{\displaystyle S}
386:{\displaystyle S}
362:{\displaystyle X}
327:{\displaystyle X}
314:with vertices in
310:The union of all
301:{\displaystyle X}
275:{\displaystyle X}
253:{\displaystyle X}
229:{\displaystyle X}
212:is defined to be
7464:
7342:(cs, lcs)-closed
7288:Effective domain
7243:Robinson–Ursescu
7119:Convex conjugate
7010:
7003:
6996:
6987:
6986:
6968:
6967:
6949:
6923:
6907:(4): 1206–1215,
6893:
6861:
6823:
6803:
6794:
6771:
6762:
6736:
6724:
6723:
6703:
6679:
6638:
6621:
6580:, translated in
6579:
6563:
6538:
6515:
6509:
6501:
6462:
6440:
6411:
6370:
6351:
6328:
6311:
6288:
6260:
6231:
6193:
6167:
6158:
6137:
6128:
6119:(6): 1689–1694,
6105:
6069:
6043:
6027:
5993:
5982:
5943:
5917:
5897:
5880:
5846:
5845:
5799:
5766:
5765:
5764:
5755:, archived from
5718:Axioms and Hulls
5713:Knuth, Donald E.
5708:
5699:
5690:(5): 2035–2053,
5676:
5651:
5649:quant-ph/0305068
5628:
5619:
5582:
5564:
5563:, E75-A: 321–329
5555:
5538:
5515:
5506:
5480:
5471:
5462:(1–3): 181–187,
5441:
5407:
5406:
5405:
5399:
5393:, archived from
5380:
5370:
5361:
5338:
5325:Convex Polytopes
5320:GrĂĽnbaum, Branko
5315:
5268:
5252:
5217:
5207:
5162:
5116:
5088:
5078:10.1090/noti2137
5072:(8): 1116–1123,
5063:
5053:
5026:
4993:
4976:
4966:Dirnböck, Hans;
4962:
4921:
4885:O'Rourke, Joseph
4881:Demaine, Erik D.
4876:
4859:
4836:
4816:
4799:
4779:
4766:
4733:
4724:
4701:
4673:Chan, Timothy M.
4668:
4644:
4614:
4573:
4548:
4517:
4500:
4470:
4454:(5–6): 265–301,
4433:
4415:10.1090/noti1810
4398:
4388:
4365:
4343:
4336:Fan, Ky (1959),
4322:
4316:
4310:
4303:
4297:
4292:, page 336, and
4283:
4277:
4268:
4262:
4256:
4250:
4244:
4238:
4232:
4226:
4220:
4214:
4208:
4202:
4196:
4190:
4181:
4175:
4169:
4163:
4157:
4151:
4145:
4139:
4133:
4127:
4121:
4115:
4109:
4103:
4097:
4091:
4085:
4079:
4073:
4067:
4061:
4055:
4049:
4043:
4037:
4031:
4025:
4019:
4010:
4004:
3998:
3992:
3986:
3980:
3974:
3968:
3962:
3956:
3950:
3944:
3938:
3932:
3929:Toussaint (1986)
3926:
3920:
3914:
3905:
3899:
3893:
3887:
3876:
3873:Toussaint (1983)
3870:
3864:
3858:
3852:
3846:
3840:
3826:
3820:
3810:
3804:
3798:
3792:
3786:
3780:
3774:
3768:
3762:
3756:
3750:
3744:
3738:
3732:
3726:
3720:
3714:
3708:
3702:
3696:
3693:Rappoport (1992)
3690:
3684:
3678:
3672:
3666:
3660:
3654:
3648:
3642:
3636:
3633:Schneider (1993)
3626:
3620:
3614:
3608:
3602:
3596:
3590:
3584:
3574:
3568:
3562:
3556:
3549:
3543:
3529:
3523:
3510:
3504:
3498:
3492:
3486:
3480:
3466:
3460:
3450:
3441:
3435:
3429:
3423:
3417:
3403:
3397:
3391:
3380:
3374:
3365:
3354:closure operator
3348:The terminology
3346:
2971:unitary elements
2932:Newton polytopes
2858:unitary elements
2814:
2812:
2811:
2806:
2801:
2800:
2789:
2776:
2774:
2773:
2768:
2766:
2765:
2760:
2707:hyperconvex hull
2679:
2677:
2676:
2671:
2666:
2644:
2642:
2641:
2636:
2624:
2622:
2621:
2616:
2563:
2549:
2533:of a point set.
2505:
2503:
2502:
2497:
2492:
2491:
2484:
2455:
2453:
2452:
2447:
2424:Chan's algorithm
2422:. These include
2421:
2419:
2418:
2413:
2379:
2377:
2376:
2371:
2341:
2339:
2338:
2333:
2317:
2315:
2314:
2309:
2297:
2295:
2294:
2289:
2277:
2275:
2274:
2269:
2249:undirected graph
2247:of the hull, an
2215:convex conjugate
2208:
2206:
2205:
2200:
2184:
2182:
2181:
2176:
2160:
2158:
2157:
2152:
2078:
2076:
2075:
2070:
2062:
2037:
2035:
2034:
2029:
2017:
2015:
2014:
2009:
1992:
1977:
1975:
1974:
1969:
1899:
1897:
1896:
1891:
1879:
1877:
1876:
1871:
1866:
1865:
1858:
1829:
1827:
1826:
1821:
1809:
1807:
1806:
1801:
1777:general position
1774:
1772:
1771:
1766:
1754:
1752:
1751:
1746:
1730:
1728:
1727:
1722:
1720:
1719:
1714:
1697:
1695:
1694:
1689:
1667:
1665:
1664:
1659:
1657:
1656:
1651:
1569:
1567:
1566:
1561:
1549:
1547:
1546:
1541:
1529:
1527:
1526:
1521:
1500:
1498:
1497:
1492:
1480:
1478:
1477:
1472:
1460:
1458:
1457:
1452:
1434:
1432:
1431:
1426:
1414:
1412:
1411:
1406:
1385:
1383:
1382:
1377:
1365:
1363:
1362:
1357:
1336:closure operator
1330:Closure operator
1273:upper half-plane
1263:
1261:
1260:
1255:
1253:
1249:
1248:
1246:
1245:
1244:
1225:
1214:
1213:
1168:upper half-plane
1149:
1147:
1146:
1141:
1122:
1120:
1119:
1114:
1102:
1100:
1099:
1094:
1079:
1077:
1076:
1071:
1059:
1057:
1056:
1051:
1028:
1026:
1025:
1020:
1004:
1002:
1001:
996:
984:
982:
981:
976:
960:
958:
957:
952:
928:open convex hull
922:of a set is the
895:
893:
892:
887:
875:
873:
872:
867:
847:
845:
844:
839:
815:
813:
812:
807:
795:
793:
792:
787:
769:
767:
766:
761:
749:
747:
746:
741:
729:
727:
726:
721:
702:
700:
699:
694:
682:
680:
679:
674:
662:
660:
659:
654:
642:
640:
639:
634:
622:
620:
619:
614:
599:
597:
596:
591:
579:
577:
576:
571:
559:
557:
556:
551:
539:
537:
536:
531:
519:
517:
516:
511:
499:
497:
496:
491:
479:
477:
476:
471:
419:obstacle problem
412:
410:
409:
404:
392:
390:
389:
384:
368:
366:
365:
360:
333:
331:
330:
325:
307:
305:
304:
299:
281:
279:
278:
273:
259:
257:
256:
251:
235:
233:
232:
227:
148:
146:
145:
140:
89:closure operator
75:Convex hulls of
7472:
7471:
7467:
7466:
7465:
7463:
7462:
7461:
7447:Convex analysis
7427:
7426:
7425:
7420:
7404:
7371:
7326:
7257:
7183:
7174:Semi-continuity
7159:Convex function
7140:Logarithmically
7107:
7068:Convex geometry
7049:
7040:Convex function
7023:
7017:Convex analysis
7014:
6934:
6931:
6926:
6913:10.2307/2533254
6843:10.2307/2046536
6815:(68): 517–526,
6721:10.1.1.155.5671
6503:
6502:
6491:
6467:Schneider, Rolf
6393:10.2307/1970292
6360:Convex Analysis
6349:
6258:
6221:
6156:10.4171/ZAA/952
6067:
5933:
5869:10.2307/1968735
5802:Hans Rademacher
5762:
5760:
5737:
5580:
5450:Herrlich, Horst
5431:
5403:
5401:
5397:
5378:
5336:
5287:Yao, F. Frances
5269:; reprinted in
5215:
5197:
5171:Kapranov, M. M.
5167:Gel'fand, I. M.
5136:10.2307/2044692
5114:
5061:
5006:Seidel, Raimund
4974:
4944:10.2307/2302604
4911:
4797:10.1.1.113.8709
4777:
4661:Schwarzkopf, O.
4653:van Kreveld, M.
4596:10.2307/1989687
4546:10.1.1.134.6921
4521:Basch, Julien;
4442:Seidel, Raimund
4396:
4331:
4326:
4325:
4317:
4313:
4304:
4300:
4284:
4280:
4269:
4265:
4257:
4253:
4245:
4241:
4233:
4229:
4221:
4217:
4209:
4205:
4197:
4193:
4182:
4178:
4170:
4166:
4158:
4154:
4146:
4142:
4134:
4130:
4122:
4118:
4110:
4106:
4098:
4094:
4086:
4082:
4074:
4070:
4062:
4058:
4050:
4046:
4038:
4034:
4028:Prasolov (2004)
4026:
4022:
4011:
4007:
3999:
3995:
3989:Chazelle (1985)
3987:
3983:
3975:
3971:
3963:
3959:
3953:Herrlich (1992)
3951:
3947:
3939:
3935:
3927:
3923:
3915:
3908:
3900:
3896:
3888:
3879:
3871:
3867:
3859:
3855:
3847:
3843:
3827:
3823:
3813:Chazelle (1993)
3811:
3807:
3799:
3795:
3787:
3783:
3775:
3771:
3763:
3759:
3751:
3747:
3739:
3735:
3727:
3723:
3715:
3711:
3703:
3699:
3691:
3687:
3679:
3675:
3667:
3663:
3657:GrĂĽnbaum (2003)
3655:
3651:
3643:
3639:
3627:
3623:
3615:
3611:
3605:Kiselman (2002)
3603:
3599:
3591:
3587:
3575:
3571:
3563:
3559:
3550:
3546:
3532:GrĂĽnbaum (2003)
3530:
3526:
3513:Steinitz (1914)
3511:
3507:
3499:
3495:
3487:
3483:
3467:
3463:
3451:
3444:
3436:
3432:
3424:
3420:
3416:, May 16, 2014.
3404:
3400:
3392:
3383:
3375:
3368:
3347:
3343:
3338:
3318:Hans Rademacher
3302:Henry Oldenburg
3294:
3285:stoichiometries
3270:
3266:
3255:Convex hull of
3249:
3229:quantum physics
3225:
3223:Quantum physics
3196:
3165:
3129:
3123:
3091:
3079:decision theory
3077:In statistical
3048:
3032:Brownian motion
2983:Radon's theorem
2955:numerical range
2924:Newton polygons
2910:
2846:
2838:polynomial time
2790:
2785:
2784:
2782:
2779:
2778:
2761:
2756:
2755:
2753:
2750:
2749:
2746:Voronoi diagram
2662:
2657:
2654:
2653:
2630:
2627:
2626:
2610:
2607:
2606:
2573:
2572:
2571:
2570:
2569:
2564:
2556:
2555:
2550:
2539:
2480:
2473:
2469:
2461:
2458:
2457:
2435:
2432:
2431:
2389:
2386:
2385:
2347:
2344:
2343:
2327:
2324:
2323:
2303:
2300:
2299:
2283:
2280:
2279:
2263:
2260:
2259:
2243:describing the
2229:
2223:
2194:
2191:
2190:
2187:convex function
2170:
2167:
2166:
2146:
2143:
2142:
2135:
2129:
2085:
2058:
2047:
2044:
2043:
2023:
2020:
2019:
1988:
1983:
1980:
1979:
1963:
1960:
1959:
1952:Brownian motion
1948:
1946:Brownian motion
1928:polygonal chain
1912:
1906:
1904:Simple polygons
1885:
1882:
1881:
1854:
1847:
1843:
1835:
1832:
1831:
1815:
1812:
1811:
1795:
1792:
1791:
1760:
1757:
1756:
1740:
1737:
1736:
1715:
1710:
1709:
1707:
1704:
1703:
1700:convex polytope
1677:
1674:
1673:
1652:
1647:
1646:
1638:
1635:
1634:
1631:
1629:Convex polytope
1617:
1612:
1604:projective dual
1600:
1584:
1555:
1552:
1551:
1535:
1532:
1531:
1515:
1512:
1511:
1486:
1483:
1482:
1466:
1463:
1462:
1440:
1437:
1436:
1420:
1417:
1416:
1400:
1397:
1396:
1371:
1368:
1367:
1351:
1348:
1347:
1332:
1327:
1302:
1296:
1271:) has the open
1269:witch of Agnesi
1240:
1236:
1229:
1224:
1209:
1208:
1192:
1188:
1186:
1183:
1182:
1164:witch of Agnesi
1156:
1132:
1129:
1128:
1108:
1105:
1104:
1085:
1082:
1081:
1065:
1062:
1061:
1045:
1042:
1041:
1014:
1011:
1010:
990:
987:
986:
970:
967:
966:
946:
943:
942:
916:
911:
902:
881:
878:
877:
861:
858:
857:
821:
818:
817:
801:
798:
797:
775:
772:
771:
755:
752:
751:
735:
732:
731:
715:
712:
711:
688:
685:
684:
668:
665:
664:
648:
645:
644:
628:
625:
624:
608:
605:
604:
585:
582:
581:
565:
562:
561:
545:
542:
541:
525:
522:
521:
505:
502:
501:
485:
482:
481:
465:
462:
461:
450:
426:bounding volume
398:
395:
394:
378:
375:
374:
354:
351:
350:
319:
316:
315:
293:
290:
289:
284:The set of all
267:
264:
263:
245:
242:
241:
221:
218:
217:
210:Euclidean space
198:
186:Voronoi diagram
162:Brownian motion
158:simple polygons
116:
113:
112:
62:Euclidean space
50:convex envelope
24:
17:
12:
11:
5:
7470:
7460:
7459:
7454:
7449:
7444:
7439:
7422:
7421:
7419:
7418:
7412:
7410:
7406:
7405:
7403:
7402:
7397:
7395:Strong duality
7392:
7387:
7381:
7379:
7373:
7372:
7370:
7369:
7334:
7332:
7328:
7327:
7325:
7324:
7319:
7310:
7305:
7303:John ellipsoid
7300:
7295:
7290:
7285:
7271:
7265:
7263:
7259:
7258:
7256:
7255:
7250:
7245:
7240:
7235:
7230:
7225:
7220:
7215:
7210:
7205:
7200:
7194:
7192:
7190:results (list)
7185:
7184:
7182:
7181:
7176:
7171:
7166:
7164:Invex function
7161:
7152:
7147:
7142:
7137:
7132:
7126:
7121:
7115:
7113:
7109:
7108:
7106:
7105:
7100:
7095:
7090:
7085:
7080:
7075:
7070:
7065:
7063:Choquet theory
7059:
7057:
7051:
7050:
7048:
7047:
7042:
7037:
7031:
7029:
7028:Basic concepts
7025:
7024:
7013:
7012:
7005:
6998:
6990:
6984:
6983:
6969:
6950:
6930:
6929:External links
6927:
6925:
6924:
6894:
6862:
6837:(2): 376–377,
6824:
6804:
6785:(2): 179–184,
6772:
6753:(2): 127–149,
6737:
6725:
6704:
6695:(1–2): 62–70,
6680:
6639:
6612:(1): 183–201,
6592:
6571:(6): 239–256,
6564:
6529:(4): 187–202,
6516:
6489:
6463:
6454:(1): 223–227,
6441:
6432:(4): 382–387,
6420:Tukey, John W.
6412:
6387:(3): 470–493,
6371:
6352:
6347:
6329:
6289:
6272:(4): 235–240,
6261:
6256:
6232:
6219:
6194:
6185:(3): 157–171,
6168:
6149:(2): 303–314,
6138:
6106:
6081:(3): 635–639,
6070:
6065:
6044:
6028:
6011:(5): 201–206,
5994:
5983:
5944:
5931:
5918:
5909:(2): 150–162,
5898:
5863:(3): 556–583,
5847:
5812:
5783:(1): 208–210,
5767:
5735:
5709:
5677:
5629:
5583:
5578:
5565:
5556:
5536:10.1.1.14.4965
5529:(2): 129–144,
5516:
5481:
5446:
5429:
5408:
5371:
5352:(4): 299–301,
5339:
5334:
5316:
5299:(4): 324–331,
5279:
5253:
5208:
5195:
5163:
5117:
5112:
5089:
5054:
5027:
5018:(4): 551–559,
4994:
4985:(2): 105–118,
4963:
4938:(4): 199–209,
4922:
4909:
4877:
4850:(1): 144–150,
4837:
4817:
4790:(1): 377–409,
4767:
4750:(4): 509–517,
4734:
4715:(2): 155–182,
4702:
4685:(4): 341–364,
4669:
4657:Overmars, Mark
4645:
4628:(5): 223–228,
4615:
4590:(2): 357–378,
4574:
4518:
4491:(1): 109–114,
4471:
4434:
4409:(3): 330–340,
4389:
4366:
4357:(5): 216–219,
4344:
4332:
4330:
4327:
4324:
4323:
4311:
4298:
4278:
4271:Hautier (2014)
4263:
4251:
4239:
4227:
4215:
4203:
4191:
4186:, p. 137–140;
4176:
4164:
4152:
4140:
4128:
4116:
4104:
4092:
4080:
4068:
4056:
4052:Gardner (1984)
4044:
4040:Johnson (1976)
4032:
4020:
4005:
3993:
3981:
3969:
3957:
3945:
3933:
3921:
3906:
3894:
3877:
3865:
3853:
3841:
3821:
3805:
3793:
3781:
3779:, p. 149.
3769:
3757:
3745:
3733:
3721:
3709:
3697:
3685:
3683:, p. 245.
3673:
3671:, p. 256.
3661:
3649:
3647:, p. 254.
3637:
3621:
3609:
3597:
3585:
3569:
3565:Whitley (1986)
3557:
3544:
3524:
3505:
3493:
3481:
3461:
3442:
3430:
3426:Oberman (2007)
3418:
3398:
3381:
3366:
3350:convex closure
3340:
3339:
3337:
3334:
3293:
3290:
3277:thermodynamics
3268:
3264:
3248:
3247:Thermodynamics
3245:
3224:
3221:
3195:
3192:
3164:
3161:
3125:Main article:
3122:
3119:
3090:
3087:
3047:
3044:
3017:triangulations
3002:ruled surfaces
2926:of univariate
2909:
2906:
2845:
2842:
2804:
2799:
2796:
2793:
2788:
2764:
2759:
2734:
2733:
2718:
2703:
2696:
2689:simple polygon
2681:
2669:
2665:
2661:
2650:
2634:
2614:
2599:
2592:
2585:
2576:For instance:
2565:
2558:
2557:
2551:
2544:
2543:
2542:
2541:
2540:
2538:
2535:
2495:
2490:
2487:
2483:
2479:
2476:
2472:
2468:
2465:
2445:
2442:
2439:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2331:
2307:
2287:
2267:
2237:representation
2225:Main article:
2222:
2219:
2198:
2174:
2150:
2141:of a function
2131:Main article:
2128:
2125:
2109:ruled surfaces
2084:
2081:
2068:
2065:
2061:
2057:
2054:
2051:
2027:
2007:
2004:
2001:
1998:
1995:
1991:
1987:
1967:
1947:
1944:
1924:simple polygon
1908:Main article:
1905:
1902:
1889:
1869:
1864:
1861:
1857:
1853:
1850:
1846:
1842:
1839:
1819:
1799:
1764:
1744:
1718:
1713:
1687:
1684:
1681:
1670:convex polygon
1655:
1650:
1645:
1642:
1627:Main article:
1616:
1613:
1611:
1608:
1599:
1596:
1583:
1580:
1572:
1571:
1559:
1539:
1519:
1502:
1490:
1470:
1450:
1447:
1444:
1424:
1404:
1392:non-decreasing
1387:
1375:
1355:
1331:
1328:
1326:
1323:
1319:Choquet theory
1298:Main article:
1295:
1294:Extreme points
1292:
1265:
1264:
1252:
1243:
1239:
1235:
1232:
1228:
1223:
1220:
1217:
1212:
1207:
1204:
1201:
1198:
1195:
1191:
1155:
1152:
1139:
1136:
1125:cross-polytope
1112:
1092:
1089:
1069:
1049:
1018:
994:
974:
950:
915:
912:
910:
907:
901:
898:
885:
865:
837:
834:
831:
828:
825:
805:
785:
782:
779:
759:
739:
719:
692:
672:
652:
632:
612:
589:
569:
549:
540:that contains
529:
509:
489:
469:
449:
446:
402:
382:
358:
335:
334:
323:
308:
297:
282:
271:
260:
249:
225:
197:
194:
138:
135:
132:
129:
126:
123:
120:
85:extreme points
54:convex closure
15:
9:
6:
4:
3:
2:
7469:
7458:
7455:
7453:
7450:
7448:
7445:
7443:
7440:
7438:
7435:
7434:
7432:
7417:
7414:
7413:
7411:
7407:
7401:
7398:
7396:
7393:
7391:
7388:
7386:
7383:
7382:
7380:
7378:
7374:
7367:
7365:
7359:
7357:
7351:
7347:
7343:
7339:
7336:
7335:
7333:
7329:
7323:
7320:
7318:
7314:
7311:
7309:
7306:
7304:
7301:
7299:
7296:
7294:
7291:
7289:
7286:
7284:
7280:
7276:
7272:
7270:
7267:
7266:
7264:
7260:
7254:
7251:
7249:
7246:
7244:
7241:
7239:
7236:
7234:
7233:Mazur's lemma
7231:
7229:
7226:
7224:
7221:
7219:
7216:
7214:
7211:
7209:
7206:
7204:
7201:
7199:
7196:
7195:
7193:
7191:
7186:
7180:
7179:Subderivative
7177:
7175:
7172:
7170:
7167:
7165:
7162:
7160:
7156:
7153:
7151:
7148:
7146:
7143:
7141:
7138:
7136:
7133:
7131:
7127:
7125:
7122:
7120:
7117:
7116:
7114:
7110:
7104:
7101:
7099:
7096:
7094:
7091:
7089:
7086:
7084:
7081:
7079:
7076:
7074:
7071:
7069:
7066:
7064:
7061:
7060:
7058:
7056:
7055:Topics (list)
7052:
7046:
7043:
7041:
7038:
7036:
7033:
7032:
7030:
7026:
7022:
7018:
7011:
7006:
7004:
6999:
6997:
6992:
6991:
6988:
6981:
6977:
6973:
6972:"Convex Hull"
6970:
6966:
6965:
6960:
6959:"Convex Hull"
6956:
6951:
6947:
6943:
6942:
6937:
6936:"Convex hull"
6933:
6932:
6922:
6918:
6914:
6910:
6906:
6902:
6901:
6895:
6892:
6888:
6884:
6880:
6876:
6872:
6868:
6863:
6860:
6856:
6852:
6848:
6844:
6840:
6836:
6832:
6831:
6825:
6822:
6818:
6814:
6810:
6805:
6802:
6798:
6793:
6788:
6784:
6780:
6779:
6773:
6770:
6766:
6761:
6756:
6752:
6748:
6747:
6742:
6738:
6734:
6730:
6726:
6722:
6717:
6713:
6709:
6705:
6702:
6698:
6694:
6690:
6686:
6681:
6678:
6674:
6670:
6666:
6662:
6658:
6655:(144): 1–40,
6654:
6650:
6649:
6644:
6640:
6637:
6633:
6629:
6625:
6620:
6615:
6611:
6607:
6606:
6601:
6597:
6593:
6591:
6587:
6583:
6578:
6574:
6570:
6565:
6562:
6558:
6554:
6550:
6546:
6542:
6537:
6532:
6528:
6524:
6523:
6517:
6513:
6507:
6500:
6496:
6492:
6490:0-521-35220-7
6486:
6482:
6478:
6474:
6473:
6468:
6464:
6461:
6457:
6453:
6449:
6448:
6442:
6439:
6435:
6431:
6427:
6426:
6421:
6418:; Ruts, Ida;
6417:
6413:
6410:
6406:
6402:
6398:
6394:
6390:
6386:
6382:
6381:
6376:
6372:
6369:
6365:
6361:
6357:
6353:
6350:
6344:
6340:
6339:
6334:
6330:
6327:
6323:
6319:
6315:
6310:
6305:
6301:
6297:
6296:
6290:
6287:
6283:
6279:
6275:
6271:
6267:
6262:
6259:
6253:
6249:
6245:
6241:
6237:
6233:
6230:
6226:
6222:
6220:3-540-40714-6
6216:
6212:
6208:
6204:
6200:
6195:
6192:
6188:
6184:
6180:
6179:
6174:
6169:
6166:
6162:
6157:
6152:
6148:
6144:
6139:
6136:
6132:
6127:
6122:
6118:
6114:
6113:
6107:
6104:
6100:
6096:
6092:
6088:
6084:
6080:
6076:
6071:
6068:
6062:
6058:
6054:
6050:
6045:
6041:
6037:
6033:
6032:Newton, Isaac
6029:
6026:
6022:
6018:
6014:
6010:
6006:
6005:
6000:
5995:
5992:, p. 698
5991:
5990:
5984:
5981:
5977:
5973:
5969:
5965:
5961:
5957:
5953:
5949:
5945:
5942:
5938:
5934:
5932:0-471-09584-2
5928:
5924:
5919:
5916:
5912:
5908:
5904:
5899:
5896:
5892:
5888:
5884:
5879:
5874:
5870:
5866:
5862:
5858:
5857:
5852:
5848:
5844:
5839:
5835:
5831:
5830:
5825:
5821:
5820:Milman, David
5817:
5813:
5811:
5807:
5803:
5798:
5794:
5790:
5786:
5782:
5778:
5777:
5772:
5768:
5759:on 2017-06-20
5758:
5754:
5750:
5746:
5742:
5738:
5736:3-540-55611-7
5732:
5728:
5724:
5720:
5719:
5714:
5710:
5707:
5703:
5698:
5693:
5689:
5685:
5684:
5678:
5675:
5671:
5667:
5663:
5659:
5655:
5650:
5645:
5642:(1): 95–102,
5641:
5637:
5636:
5630:
5627:
5623:
5618:
5613:
5609:
5605:
5601:
5597:
5593:
5589:
5584:
5581:
5579:9780080540221
5575:
5571:
5566:
5562:
5557:
5554:
5550:
5546:
5542:
5537:
5532:
5528:
5524:
5523:
5517:
5514:
5510:
5505:
5500:
5496:
5492:
5491:
5486:
5482:
5479:
5475:
5470:
5465:
5461:
5457:
5456:
5451:
5447:
5445:
5440:
5436:
5432:
5426:
5422:
5418:
5414:
5409:
5400:on 2021-02-28
5396:
5392:
5388:
5384:
5377:
5372:
5369:
5365:
5360:
5355:
5351:
5347:
5346:
5340:
5337:
5335:9780387004242
5331:
5327:
5326:
5321:
5317:
5314:
5310:
5306:
5302:
5298:
5294:
5293:
5288:
5284:
5280:
5278:
5274:
5273:
5266:
5262:
5258:
5254:
5251:
5247:
5243:
5239:
5235:
5231:
5227:
5223:
5222:
5214:
5209:
5206:
5202:
5198:
5196:0-8176-3660-9
5192:
5188:
5184:
5180:
5176:
5172:
5168:
5164:
5161:
5157:
5153:
5149:
5145:
5141:
5137:
5133:
5129:
5125:
5124:
5118:
5115:
5113:9781108641449
5109:
5105:
5101:
5097:
5096:
5090:
5087:
5083:
5079:
5075:
5071:
5067:
5060:
5055:
5052:
5048:
5044:
5040:
5036:
5032:
5028:
5025:
5021:
5017:
5013:
5012:
5007:
5003:
4999:
4995:
4992:
4988:
4984:
4980:
4973:
4969:
4964:
4961:
4957:
4953:
4949:
4945:
4941:
4937:
4933:
4932:
4927:
4923:
4920:
4916:
4912:
4906:
4902:
4898:
4894:
4890:
4886:
4882:
4878:
4875:
4871:
4867:
4863:
4858:
4853:
4849:
4845:
4844:
4838:
4835:
4831:
4827:
4823:
4818:
4815:
4811:
4807:
4803:
4798:
4793:
4789:
4785:
4784:
4776:
4772:
4768:
4765:
4761:
4757:
4753:
4749:
4745:
4744:
4739:
4735:
4732:
4728:
4723:
4718:
4714:
4710:
4709:
4703:
4700:
4696:
4692:
4688:
4684:
4680:
4679:
4674:
4670:
4666:
4662:
4658:
4654:
4650:
4646:
4643:
4639:
4635:
4631:
4627:
4623:
4622:
4616:
4613:
4609:
4605:
4601:
4597:
4593:
4589:
4585:
4584:
4579:
4575:
4572:
4568:
4564:
4560:
4556:
4552:
4547:
4542:
4538:
4534:
4533:
4528:
4524:
4519:
4516:
4512:
4508:
4504:
4499:
4494:
4490:
4486:
4485:
4480:
4476:
4472:
4469:
4465:
4461:
4457:
4453:
4449:
4448:
4443:
4439:
4435:
4432:
4428:
4424:
4420:
4416:
4412:
4408:
4404:
4403:
4395:
4390:
4387:
4383:
4379:
4375:
4371:
4367:
4364:
4360:
4356:
4352:
4351:
4345:
4341:
4340:
4334:
4333:
4320:
4315:
4308:
4302:
4295:
4291:
4287:
4286:Newton (1676)
4282:
4276:
4272:
4267:
4260:
4255:
4248:
4243:
4236:
4231:
4224:
4219:
4212:
4207:
4200:
4199:Worton (1995)
4195:
4189:
4185:
4180:
4173:
4168:
4161:
4156:
4149:
4148:Nicola (2000)
4144:
4137:
4132:
4125:
4120:
4113:
4112:Harris (1971)
4108:
4101:
4096:
4089:
4084:
4077:
4072:
4065:
4060:
4053:
4048:
4041:
4036:
4029:
4024:
4018:
4014:
4009:
4002:
3997:
3990:
3985:
3978:
3973:
3966:
3961:
3954:
3949:
3942:
3937:
3930:
3925:
3918:
3913:
3911:
3903:
3898:
3891:
3886:
3884:
3882:
3874:
3869:
3862:
3857:
3850:
3845:
3838:
3834:
3830:
3825:
3818:
3814:
3809:
3803:, p. 13.
3802:
3797:
3790:
3785:
3778:
3773:
3767:, p. 36.
3766:
3761:
3754:
3753:Seaton (2017)
3749:
3742:
3737:
3730:
3729:Sedykh (1981)
3725:
3718:
3713:
3706:
3701:
3694:
3689:
3682:
3677:
3670:
3665:
3659:, p. 57.
3658:
3653:
3646:
3641:
3634:
3630:
3625:
3618:
3613:
3606:
3601:
3594:
3589:
3582:
3578:
3573:
3566:
3561:
3555:, Remark 2.6.
3554:
3553:Talman (1977)
3548:
3541:
3540:Sakuma (1977)
3537:
3533:
3528:
3522:
3518:
3517:Gustin (1947)
3514:
3509:
3503:, p. 99.
3502:
3497:
3490:
3489:Sontag (1982)
3485:
3478:
3477:Andrew (1979)
3474:
3470:
3465:
3458:
3454:
3449:
3447:
3439:
3434:
3427:
3422:
3415:
3411:
3407:
3402:
3395:
3390:
3388:
3386:
3379:, p. 12.
3378:
3373:
3371:
3363:
3359:
3355:
3351:
3345:
3341:
3333:
3331:
3327:
3323:
3320:'s review of
3319:
3315:
3311:
3307:
3303:
3299:
3289:
3286:
3282:
3278:
3263:compounds. Mg
3262:
3258:
3253:
3244:
3242:
3238:
3234:
3230:
3220:
3218:
3217:neighborhoods
3214:
3209:
3205:
3201:
3191:
3189:
3185:
3181:
3176:
3174:
3170:
3160:
3158:
3154:
3150:
3146:
3142:
3138:
3134:
3128:
3118:
3116:
3112:
3108:
3104:
3100:
3096:
3086:
3084:
3080:
3075:
3073:
3069:
3065:
3057:
3052:
3043:
3041:
3037:
3033:
3028:
3026:
3022:
3018:
3015:
3011:
3007:
3003:
2999:
2995:
2990:
2988:
2984:
2980:
2976:
2972:
2968:
2964:
2960:
2959:normal matrix
2956:
2952:
2947:
2945:
2941:
2937:
2933:
2929:
2925:
2919:
2914:
2905:
2903:
2899:
2898:BĂ©zier curves
2895:
2891:
2887:
2883:
2879:
2875:
2871:
2867:
2863:
2859:
2855:
2851:
2841:
2839:
2835:
2830:
2827:
2826:convex layers
2823:
2818:
2802:
2797:
2794:
2791:
2762:
2747:
2743:
2739:
2731:
2727:
2723:
2719:
2716:
2712:
2708:
2704:
2701:
2697:
2694:
2690:
2686:
2682:
2667:
2663:
2659:
2651:
2648:
2632:
2612:
2604:
2600:
2597:
2593:
2590:
2586:
2583:
2579:
2578:
2577:
2568:
2562:
2554:
2548:
2534:
2532:
2528:
2524:
2519:
2515:
2511:
2509:
2485:
2481:
2477:
2470:
2463:
2443:
2440:
2437:
2429:
2425:
2406:
2403:
2400:
2397:
2391:
2383:
2364:
2361:
2358:
2355:
2349:
2329:
2321:
2305:
2285:
2265:
2256:
2254:
2250:
2246:
2242:
2238:
2234:
2228:
2218:
2216:
2212:
2196:
2189:majorized by
2188:
2172:
2164:
2148:
2140:
2134:
2124:
2122:
2118:
2114:
2110:
2106:
2102:
2094:
2089:
2080:
2066:
2063:
2059:
2055:
2052:
2049:
2041:
2025:
2005:
2002:
1999:
1996:
1993:
1989:
1985:
1978:in the range
1965:
1957:
1953:
1943:
1941:
1937:
1933:
1929:
1925:
1916:
1911:
1901:
1887:
1859:
1855:
1851:
1844:
1837:
1817:
1797:
1789:
1784:
1782:
1778:
1762:
1742:
1734:
1716:
1701:
1685:
1682:
1679:
1671:
1653:
1643:
1640:
1630:
1621:
1610:Special cases
1607:
1605:
1595:
1593:
1589:
1588:Minkowski sum
1582:Minkowski sum
1579:
1577:
1557:
1537:
1517:
1509:
1508:
1503:
1488:
1468:
1448:
1445:
1442:
1422:
1402:
1394:
1393:
1388:
1373:
1353:
1345:
1341:
1340:
1339:
1337:
1322:
1320:
1316:
1312:
1307:
1306:extreme point
1301:
1291:
1289:
1288:weak topology
1285:
1281:
1276:
1274:
1270:
1250:
1241:
1237:
1233:
1230:
1226:
1221:
1218:
1215:
1202:
1199:
1196:
1189:
1181:
1180:
1179:
1177:
1169:
1165:
1160:
1151:
1137:
1134:
1126:
1110:
1090:
1087:
1067:
1047:
1038:
1036:
1032:
1016:
1008:
1005:is already a
992:
972:
964:
948:
939:
937:
933:
929:
925:
921:
906:
897:
883:
863:
855:
851:
832:
829:
826:
803:
783:
780:
777:
757:
737:
717:
709:
704:
690:
670:
650:
630:
610:
601:
587:
567:
547:
527:
507:
487:
467:
454:
445:
443:
439:
438:affine spaces
435:
431:
427:
422:
420:
416:
415:spanning tree
400:
380:
372:
356:
348:
345:with minimum
344:
340:
321:
313:
309:
295:
288:of points in
287:
283:
269:
261:
247:
239:
238:
237:
223:
215:
211:
202:
193:
191:
187:
183:
179:
178:convex layers
175:
171:
167:
163:
159:
154:
152:
133:
130:
127:
124:
118:
110:
106:
102:
98:
94:
90:
86:
82:
78:
73:
71:
67:
63:
59:
55:
51:
47:
43:
35:
30:
26:
22:
7437:Convex hulls
7400:Weak duality
7363:
7355:
7275:Orthogonally
7268:
6962:
6939:
6904:
6898:
6866:
6834:
6828:
6812:
6808:
6782:
6776:
6750:
6744:
6732:
6711:
6692:
6688:
6652:
6646:
6643:Steinitz, E.
6609:
6603:
6581:
6568:
6526:
6520:
6471:
6451:
6445:
6429:
6423:
6384:
6378:
6359:
6337:
6299:
6293:
6269:
6265:
6239:
6202:
6182:
6176:
6173:Wood, Derick
6146:
6142:
6116:
6110:
6078:
6074:
6048:
6039:
6008:
6002:
5988:
5958:(2): 87–98,
5955:
5951:
5922:
5906:
5902:
5860:
5854:
5833:
5827:
5780:
5774:
5771:KĹ‘nig, DĂ©nes
5761:, retrieved
5757:the original
5717:
5687:
5681:
5639:
5633:
5594:(1): 20253,
5591:
5587:
5569:
5560:
5526:
5520:
5497:(1): 89–94,
5494:
5488:
5459:
5453:
5412:
5402:, retrieved
5395:the original
5382:
5349:
5343:
5324:
5296:
5290:
5270:
5264:
5260:
5225:
5219:
5178:
5127:
5121:
5094:
5069:
5065:
5042:
5015:
5009:
4982:
4978:
4935:
4929:
4926:Dines, L. L.
4892:
4847:
4841:
4828:(1): 29–39,
4825:
4821:
4787:
4781:
4747:
4741:
4712:
4706:
4682:
4676:
4664:
4625:
4619:
4587:
4581:
4536:
4530:
4488:
4482:
4475:Bárány, Imre
4451:
4445:
4406:
4400:
4377:
4354:
4348:
4338:
4319:Dines (1938)
4314:
4307:White (1923)
4301:
4281:
4275:Fultz (2020)
4266:
4259:Gibbs (1873)
4254:
4242:
4230:
4218:
4206:
4194:
4179:
4172:Mason (1908)
4167:
4155:
4143:
4136:Katoh (1992)
4131:
4119:
4107:
4095:
4088:Weeks (1993)
4083:
4071:
4059:
4047:
4035:
4023:
4013:Artin (1967)
4008:
3996:
3984:
3977:Brown (1979)
3972:
3965:Rossi (1961)
3960:
3948:
3936:
3924:
3897:
3868:
3856:
3844:
3824:
3808:
3796:
3784:
3772:
3760:
3748:
3736:
3724:
3712:
3700:
3688:
3676:
3664:
3652:
3640:
3624:
3612:
3600:
3588:
3572:
3560:
3547:
3527:
3508:
3496:
3484:
3464:
3438:Knuth (1992)
3433:
3421:
3414:MathOverflow
3401:
3396:, p. 3.
3357:
3349:
3344:
3298:Isaac Newton
3295:
3274:
3226:
3197:
3177:
3173:BĂ©zier curve
3166:
3130:
3092:
3076:
3061:
3036:space curves
3029:
2998:ideal points
2991:
2948:
2922:
2847:
2844:Applications
2834:convex skull
2831:
2822:circumradius
2817:alpha shapes
2735:
2715:metric space
2596:conical hull
2574:
2512:
2257:
2253:face lattice
2230:
2136:
2098:
2083:Space curves
1949:
1935:
1931:
1921:
1785:
1632:
1601:
1585:
1573:
1505:
1390:
1343:
1333:
1303:
1284:Banach space
1277:
1266:
1173:
1039:
940:
927:
919:
917:
903:
705:
602:
480:, for every
459:
423:
339:bounded sets
336:
207:
190:convex skull
166:space curves
155:
91:, and every
81:compact sets
74:
53:
49:
45:
39:
25:
7390:Duality gap
7385:Dual system
7269:Convex hull
6375:Rossi, Hugo
6203:Polynomials
5999:Avis, David
5836:: 133–138,
5816:Krein, Mark
4649:de Berg, M.
4539:(1): 1–28,
4479:Pach, János
4438:Avis, David
4370:Artin, Emil
4309:, page 520.
4305:See, e.g.,
4290:Auel (2019)
4064:Reay (1979)
3849:Chan (2012)
3593:Okon (2000)
3473:Graham scan
3330:Lloyd Dines
3233:state space
3219:of points.
3180:chain girth
3141:budget sets
3072:Tukey depth
2963:eigenvalues
2928:polynomials
2908:Mathematics
2870:Tukey depth
2854:eigenvalues
2850:polynomials
2603:visual hull
2589:linear hull
2582:affine hull
2508:linear time
2320:Graham scan
2221:Computation
2217:operation.
2105:developable
2101:space curve
1576:antimatroid
1035:compact set
965:containing
963:half-spaces
371:rubber band
349:containing
196:Definitions
105:half-spaces
97:algorithmic
93:antimatroid
46:convex hull
7431:Categories
7313:Radial set
7283:Convex set
7045:Convex set
6900:Biometrics
6536:1603.08409
5948:Lee, D. T.
5810:48.0835.01
5763:2011-09-15
5404:2020-01-01
5130:(1): 171,
5035:Marden, A.
4329:References
3837:Lee (1983)
3581:Lay (1982)
3536:Lay (1982)
3457:Lay (1982)
3362:Fan (1959)
3204:home range
3188:skin girth
3153:non-convex
3046:Statistics
2975:C*-algebra
2936:asymptotic
2902:home range
1810:points in
1507:idempotent
1103:points of
1031:finite set
1007:closed set
796:points in
560:, because
58:convex set
34:convex set
7298:Hypograph
6964:MathWorld
6946:EMS Press
6883:1853/3736
6716:CiteSeerX
6677:122998337
6326:121352925
5851:Krein, M.
5797:128041360
5531:CiteSeerX
5277:pp. 33–54
5267:: 382–404
5221:Ecography
5160:119501393
5086:221659506
4792:CiteSeerX
4541:CiteSeerX
3819:, p. 256.
3538:, p. 21;
3534:, p. 16;
3455:, p. 12;
3257:magnesium
3121:Economics
3014:canonical
2852:, matrix
2668:α
2489:⌋
2475:⌊
2404:
2362:
2127:Functions
2117:sphericon
2067:θ
2056:π
2053:−
2026:θ
2006:π
2000:θ
1986:π
1966:θ
1863:⌋
1849:⌊
1644:⊂
1446:⊆
1344:extensive
1222:≥
1216:
347:perimeter
312:simplices
131:
77:open sets
7322:Zonotope
7293:Epigraph
6891:15514388
6821:43432008
6636:18446330
6598:(1982),
6561:84179479
6506:citation
6469:(1993),
6358:(1970),
6286:20137707
6103:30843551
5980:28600832
5822:(1940),
5804:(1922),
5715:(1992),
5674:15995449
5626:31882982
5439:24287952
5322:(2003),
5250:14592779
4970:(1997),
4814:26605267
4773:(1993),
4663:(2008),
4642:44537056
4431:76650751
4372:(1967),
3583:, p. 43.
3459:, p. 17.
3208:Outliers
3200:ethology
3194:Ethology
2693:geodesic
2531:diameter
2426:and the
2163:epigraph
1668:forms a
1176:open set
932:interior
854:triangle
42:geometry
7377:Duality
7279:Pseudo-
7253:Ursescu
7150:Pseudo-
7124:Concave
7103:Simplex
7083:Duality
6982:, 2007.
6948:, 2001
6921:2533254
6859:0835903
6851:2046536
6801:0404097
6769:1241189
6701:0463985
6669:1580890
6628:0644949
6577:0630708
6553:3765242
6499:1216521
6409:0133479
6401:1970292
6368:0274683
6318:0570883
6229:2082772
6165:1768994
6135:2286077
6083:Bibcode
6025:0552534
5972:0724699
5941:0655598
5895:0002009
5887:1968735
5753:5452191
5745:1226891
5706:1881029
5654:Bibcode
5617:6934831
5596:Bibcode
5553:2107032
5513:0460358
5478:1173256
5391:0356305
5368:0020800
5313:0729228
5230:Bibcode
5205:1264417
5152:0722439
5144:2044692
5051:0903852
5041:(ed.),
4991:1622664
4960:1524247
4952:2302604
4919:2405683
4874:0972777
4866:2244202
4764:0798557
4731:0834056
4699:2994585
4612:1501815
4604:1989687
4571:8013433
4563:1670903
4515:0663877
4507:2044407
4468:1447243
4423:3889348
4386:0237460
3364:, p.48.
3324: (
3308: (
3292:History
3131:In the
3068:bagplot
3056:bagplot
2981:, both
2874:bagplot
1932:pockets
930:is the
924:closure
850:simplex
70:bounded
7360:, and
7331:Series
7248:Simons
7155:Quasi-
7145:Proper
7130:Closed
6919:
6889:
6857:
6849:
6819:
6799:
6767:
6718:
6699:
6675:
6667:
6634:
6626:
6575:
6559:
6551:
6497:
6487:
6407:
6399:
6366:
6345:
6324:
6316:
6284:
6254:
6227:
6217:
6163:
6133:
6101:
6063:
6023:
5978:
5970:
5939:
5929:
5893:
5885:
5808:
5795:
5751:
5743:
5733:
5704:
5672:
5624:
5614:
5576:
5551:
5533:
5511:
5476:
5444:p. 143
5442:; see
5437:
5427:
5389:
5366:
5332:
5311:
5248:
5203:
5193:
5158:
5150:
5142:
5110:
5084:
5049:
4989:
4958:
4950:
4917:
4907:
4872:
4864:
4812:
4794:
4762:
4729:
4697:
4640:
4610:
4602:
4569:
4561:
4543:
4513:
4505:
4466:
4429:
4421:
4384:
4288:; see
3314:German
3261:carbon
3231:, the
2965:. The
2953:, the
2856:, and
2744:, the
2245:facets
2038:. The
1733:vertex
1504:It is
1389:It is
1342:It is
214:convex
188:, and
168:, and
44:, the
7188:Main
6917:JSTOR
6887:S2CID
6847:JSTOR
6817:JSTOR
6673:S2CID
6632:S2CID
6557:S2CID
6531:arXiv
6397:JSTOR
6322:S2CID
6282:S2CID
6099:S2CID
5976:S2CID
5883:JSTOR
5793:S2CID
5749:S2CID
5670:S2CID
5644:arXiv
5398:(PDF)
5379:(PDF)
5246:S2CID
5216:(PDF)
5156:S2CID
5140:JSTOR
5082:S2CID
5062:(PDF)
4975:(PDF)
4948:JSTOR
4862:JSTOR
4810:S2CID
4778:(PDF)
4638:S2CID
4600:JSTOR
4567:S2CID
4503:JSTOR
4427:S2CID
4397:(PDF)
3336:Notes
3322:KĹ‘nig
3025:knots
2977:. In
2973:in a
2957:of a
2944:roots
2527:width
2113:oloid
2093:oloid
1672:when
1435:with
1029:is a
710:, if
7308:Lens
7262:Sets
7112:Maps
7019:and
6653:1914
6512:link
6485:ISBN
6343:ISBN
6252:ISBN
6215:ISBN
6061:ISBN
5927:ISBN
5731:ISBN
5622:PMID
5574:ISBN
5435:PMID
5425:ISBN
5330:ISBN
5191:ISBN
5108:ISBN
4905:ISBN
3326:1922
3310:1935
3184:hull
3143:and
3097:and
2985:and
2930:and
2888:and
2832:The
2815:The
2742:dual
2736:The
2720:The
2698:The
2683:The
2601:The
2594:The
2587:The
2580:The
2529:and
2441:>
2107:and
2003:<
1997:<
1602:The
1415:and
1162:The
918:The
337:For
184:and
101:dual
7362:(Hw
6974:by
6909:doi
6879:hdl
6871:doi
6839:doi
6787:doi
6755:doi
6657:doi
6614:doi
6586:doi
6541:doi
6477:doi
6456:doi
6434:doi
6389:doi
6304:doi
6274:doi
6244:doi
6207:doi
6187:doi
6151:doi
6121:doi
6117:135
6091:doi
6053:doi
6013:doi
5960:doi
5911:doi
5873:hdl
5865:doi
5838:doi
5806:JFM
5785:doi
5723:doi
5692:doi
5688:354
5662:doi
5612:PMC
5604:doi
5541:doi
5499:doi
5464:doi
5417:doi
5354:doi
5301:doi
5238:doi
5183:doi
5132:doi
5100:doi
5074:doi
5020:doi
4940:doi
4897:doi
4852:doi
4830:doi
4802:doi
4752:doi
4717:doi
4687:doi
4630:doi
4592:doi
4551:doi
4493:doi
4456:doi
4411:doi
4359:doi
3475:by
3300:to
3227:In
3167:In
3135:of
3093:In
3062:In
3019:of
3008:in
2949:In
2401:log
2359:log
2231:In
2091:An
1702:in
1304:An
1060:is
436:or
128:log
52:or
40:In
7433::
7354:(H
7352:,
7348:,
7344:,
7281:)
7277:,
7157:)
7135:K-
6978:,
6961:,
6957:,
6944:,
6938:,
6915:,
6905:51
6903:,
6885:,
6877:,
6855:MR
6853:,
6845:,
6835:97
6833:,
6813:17
6811:,
6797:MR
6795:,
6783:38
6781:,
6765:MR
6763:,
6751:52
6749:,
6714:,
6697:MR
6693:29
6691:,
6687:,
6671:,
6665:MR
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6651:,
6630:,
6624:MR
6622:,
6610:98
6608:,
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6573:MR
6555:,
6549:MR
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6539:,
6527:11
6525:,
6508:}}
6504:{{
6495:MR
6493:,
6452:14
6450:,
6430:53
6428:,
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6364:MR
6320:,
6314:MR
6312:,
6300:34
6298:,
6280:,
6270:11
6268:,
6250:,
6225:MR
6223:,
6213:,
6201:,
6183:33
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6161:MR
6159:,
6147:19
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6097:,
6089:,
6079:23
6077:,
6059:,
6038:,
6021:MR
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5974:,
5968:MR
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5935:,
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5861:41
5832:,
5826:,
5818:;
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5781:14
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5739:,
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5668:,
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5285:;
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5000:;
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3880:^
3835:;
3831:;
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3445:^
3412:,
3384:^
3369:^
3054:A
3042:.
3027:.
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2510:.
2079:.
1900:.
1783:.
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192:.
180:,
176:,
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7364:x
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7315:/
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