3315:
31:
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3250:, which is the set of all admissible positions of the object. In the simple model of translational motion of an object in the plane, where the position of an object may be uniquely specified by the position of a fixed point of this object, the configuration space are the Minkowski sum of the set of obstacles and the movable object placed at the origin and rotated 180 degrees.
3318:
Minkowski addition and convex hulls. The sixteen dark-red points (on the right) form the
Minkowski sum of the four non-convex sets (on the left), each of which consists of a pair of red points. Their convex hulls (shaded pink) contain plus-signs (+): The right plus-sign is the sum of the left
616:
An alternative definition of the
Minkowski difference is sometimes used for computing intersection of convex shapes. This is not equivalent to the previous definition, and is not an inverse of the sum operation. Instead it replaces the vector addition of the Minkowski sum with a
3368:) by a very simple procedure, which may be informally described as follows. Assume that the edges of a polygon are given and the direction, say, counterclockwise, along the polygon boundary. Then it is easily seen that these edges of the convex polygon are ordered by
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is not necessarily equivalent. The sum can fill gaps which the difference may not re-open, and the difference can erase small islands which the sum cannot recreate from nothing.
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4804:. Advanced textbooks in economics. Vol. 12 (reprint of (1971) San Francisco, CA: Holden-Day, Inc. Mathematical economics texts.
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3115:
Minkowski sums act linearly on the perimeter of two-dimensional convex bodies: the perimeter of the sum equals the sum of perimeters. Additionally, if
989:
5167:
366:
This definition allows a symmetrical relationship between the
Minkowski sum and difference. Note that alternately taking the sum and difference with
30:
5142:
2309:
is not necessarily a closed set. However, the
Minkowski sum of two closed subsets will be a closed subset if at least one of these sets is also a
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5795:
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is again positive homogeneous and convex and hence the support function of a compact convex set. This definition is fundamental in the
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which can be moved freely while keeping them parallel to their original direction. Assemble these arrows in the order of the sequence
5113:
6190:
5912:
5767:
3286:
Minkowski sums are also frequently used in aggregation theory when individual objects to be aggregated are characterized via sets.
4853:
Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In
6658:
6160:
6099:
5743:
5271:
4609:. Encyclopedia of mathematics and its applications. Vol. 44. Cambridge: Cambridge University Press. pp. xiv+490.
4573:. Encyclopedia of mathematics and its applications. Vol. 44. Cambridge: Cambridge University Press. pp. xiv+490.
1956:
is the empty set). The
Minkowski sum of a closed ball and an open ball is an open ball. More generally, the Minkowski sum of an
534:
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240:
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5363:
4933:
4907:
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4614:
4578:
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4974:
3958:{\displaystyle 1_{A\,+_{\mathrm {e} }\,B}(z)=\mathop {\mathrm {ess\,sup} } _{x\,\in \,\mathbb {R} ^{n}}1_{A}(x)1_{B}(z-x),}
1970:
4160:
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4557:, see Theorem 1.1.2 (pages 2–3) in Schneider; this reference discusses much of the literature on the
4458:
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1304:
900:
811:
6854:
6834:
6809:
6643:
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5790:
5155:
Demonstration of
Minkowski additivity, convex monotonicity, and other properties of the Earth Movers distance
5084:
3247:
2483:{\displaystyle \operatorname {Conv} (S_{1}+S_{2})=\operatorname {Conv} (S_{1})+\operatorname {Conv} (S_{2}).}
17:
6694:
6073:
5846:
5780:
5608:
3396:
by attaching the tail of the next arrow to the head of the previous arrow. It turns out that the resulting
1294:
is important in
Minkowski addition, because the empty set annihilates every other subset: for every subset
5416:
4030:
2383:
of a real vector space, the convex hull of their
Minkowski sum is the Minkowski sum of their convex hulls:
6515:
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5079:
4278:
2707:
2046:
1615:
5074:
4902:(Corrected reprint of 1965 Wiley ed.), Huntington, New York: Robert E. Krieger Publishing Company,
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421:
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779:
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5421:
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4912:– via www.krieger-publishing.com/subcats/MathematicsandStatistics/mathematicsandstatistics.html
4332:
3227:
2995:
2616:
1674:
4864:. Handbooks in economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 15–52.
6824:
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6308:
6303:
6175:
6115:
5902:
5800:
5703:
5401:
5386:
5228:
5118:
4379:"Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt"
3278:
Minkowski sums are used to outline a shape with another shape creating a composite of both shapes.
2743:
1817:
1395:
1342:
3162:
1547:
1370:
721:{\displaystyle A-B=\{\mathbf {a} -\mathbf {b} \,|\,\mathbf {a} \in A,\ \mathbf {b} \in B\}=A+(-B)}
6581:
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4291: – Integral expressing the amount of overlap of one function as it is shifted over another
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5527:
5523:
5519:
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5511:
2936:
2877:
2568:{\textstyle \operatorname {Conv} \left(\sum {S_{n}}\right)=\sum \operatorname {Conv} (S_{n}).}
1569:
6739:
6426:
6408:
6373:
6213:
5737:
5323:
4924:. Princeton landmarks in mathematics (Reprint of the 1979 Princeton mathematical series
4294:
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1791:
606:
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3416:
If one polygon is convex and another one is not, the complexity of their
Minkowski sum is O(
1205:
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5159:
4943:
4887:
4823:
4624:
4588:
4540:
4511:; Ĺ mulian, V. (1940). "On regularly convex sets in the space conjugate to a Banach space".
4300:
4245:
3656:{\displaystyle A+_{\mathrm {e} }B=\left\{z\in \mathbb {R} ^{n}\,|\,\mu \left>0\right\},}
3223:
3207:
3187:
2359:
2332:
1892:
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156:{\displaystyle A+B=\{\mathbf {a} +\mathbf {b} \,|\,\mathbf {a} \in A,\ \mathbf {b} \in B\}}
5600:
5148:
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3262:
machining, the programming of the NC tool exploits the fact that the
Minkowski sum of the
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768:, each consisting of three position vectors (informally, three points), representing the
3530:{\displaystyle A+B=\left\{z\in \mathbb {R} ^{n}\,|\,A\cap (z-B)\neq \emptyset \right\}.}
6623:
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5019:"Aggregation of Malmquist productivity indexes allowing for reallocation of resources"
4869:
4728:"Aggregation of Malmquist productivity indexes allowing for reallocation of resources"
1339:
For another example, consider the Minkowski sums of open or closed balls in the field
6335:
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5308:
5102:
4929:
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4809:
4681:
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2278:{\displaystyle G+Y=\{(x,y)\in \mathbb {R} ^{2}:x\neq 0\}=\mathbb {R} ^{2}\setminus Y}
769:
732:
47:
4487:
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4021:
3799:{\displaystyle 1_{A\,+\,B}(z)=\sup _{x\,\in \,\mathbb {R} ^{n}}1_{A}(x)1_{B}(z-x),}
3675:
1236:
602:
6611:
6150:
5668:
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5851:
5805:
5753:
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4939:
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3243:
3211:
1185:{\displaystyle A+B=\{(1,0),(2,1),(2,-1),(0,1),(1,2),(1,0),(0,-1),(1,0),(1,-2)\},}
621:. If the two convex shapes intersect, the resulting set will contain the origin.
66:
62:
51:
5678:
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6040:
5892:
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4341: – set of all possible sums of an element of set A and an element of set B
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4672:
4655:
2493:
This result holds more generally for any finite collection of non-empty sets:
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3440:
of two subsets of Euclidean space. The usual Minkowski sum can be written as
3294:
Minkowski sums, specifically Minkowski differences, are often used alongside
3186:
rotation is a disk. These two facts can be combined to give a short proof of
2587:
2310:
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4422:
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6663:
6541:
6388:
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4928: ed.). Princeton, NJ: Princeton University Press. pp. xviii+451.
4641:
4312:
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1916:
been defined to be the open ball, rather than the closed ball, centered at
4900:
Addition Theorems: The Addition Theorems of Group Theory and Number Theory
6723:
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Minkowski addition behaves well with respect to the operation of taking
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6155:
5897:
5729:
5486:
5456:
5218:
4895:
4797:
4532:
4508:
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4394:
3420:). If both of them are nonconvex, their Minkowski sum complexity is O((
3246:
of an object among obstacles. They are used for the computation of the
2814:
The figure to the right shows an example of a non-convex set for which
2306:
2181:
27:
Sums vector sets A and B by adding each vector in A to each vector in B
6688:
6505:
1291:
5154:
4524:
6653:
6648:
6606:
6586:
6556:
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4424:
GPU-Based Computation of Voxelized Minkowski Sums with Applications
4353:
3678:. The reason for the term "essential" is the following property of
3275:
3215:
2185:
1957:
1936:(the non-zero assumption is needed because the open ball of radius
773:
743:
39:
4972:(2006), "Minkowski Sums of Monotone and General Simple Polygons",
6596:
5276:
4565:
in its "Chapter 3 Minkowski addition" (pages 126–196):
4953:
Additive Number Theory: Inverse Problems and Geometry of Sumsets
4309: – Method for bounding the errors of numerical computations
3266:
with its trajectory gives the shape of the cut in the material.
4562:
4338:
3400:
will in fact be a convex polygon which is the Minkowski sum of
3352:
vertices, their Minkowski sum is a convex polygon with at most
2769:
1889:
are all non-zero then the same equalities would still hold had
4831:
Gardner, Richard J. (2002), "The Brunn-Minkowski inequality",
34:
The red figure is the Minkowski sum of blue and green figures.
3389:
5189:
5630:
1195:
which comprises the vertices of a hexagon and its center .
5098:
On the tendency toward convexity of the vector sum of sets
3309:
591:{\displaystyle A+B=(A^{\complement }-(-B))^{\complement }}
523:{\displaystyle A-B=(A^{\complement }+(-B))^{\complement }}
356:{\displaystyle A-B=(A^{\complement }+(-B))^{\complement }}
288:{\displaystyle -B=\{\mathbf {-b} \,|\,\mathbf {b} \in B\}}
3226:. It has also been shown to be closely connected to the
5143:
Wikibooks:OpenSCAD User Manual/Transformations#minkowski
1298:
of a vector space, its sum with the empty set is empty:
4656:"Properties of the d-dimensional earth mover's problem"
4358:
Pages displaying short descriptions of redirect targets
4329: – Generalization of the concept of parallel lines
5101:, Cowles Foundation discussion papers, vol. 538,
2502:
1479:{\displaystyle B_{r}:=\{s\in \mathbb {K} :|s|\leq r\}}
4859:
Handbook of mathematical economics, Volume
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2032:{\displaystyle G=\{(x,1/x):0\neq x\in \mathbb {R} \}}
1973:
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992:
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379:
302:
243:
188:
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6114:
4343:
Pages displaying wikidata descriptions as a fallback
4283:
Pages displaying wikidata descriptions as a fallback
4234:{\displaystyle h_{K+_{p}L}^{p}=h_{K}^{p}+h_{L}^{p}.}
3328:
3253:
5960:Spectral theory of ordinary differential equations
4451:"Spatial Planning: A Configuration Space Approach"
4356: – Convex polyhedron projected from hypercube
4303: – Basic operation in mathematical morphology
4233:
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4335: – Sums of sets of vectors are nearly convex
2316:
6801:
4553:For the commutativity of Minkowski addition and
4285:, an inequality on the volumes of Minkowski sums
4275: – Polytope combining two smaller polytopes
3723:
2694:{\displaystyle \mu S+\lambda S=(\mu +\lambda )S}
2305:-axis. This shows that the Minkowski sum of two
4350: – Vector space with a notion of nearness
3190:on the perimeter of curves of constant width.
605:the Minkowski sum and difference are known as
6100:
5616:
5175:
5149:Application of Minkowski Addition to robotics
5016:
4725:
4507:
4448:
4297: – Operation in mathematical morphology
4020:, the Minkowski sum can be described by the
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2206:
2104:
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1980:
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282:
253:
150:
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5103:Cowles Foundation for Research in Economics
4916:
4852:
3198:Minkowski addition plays a central role in
2740:" holds for all non-negative real numbers,
6107:
6093:
5623:
5609:
5182:
5168:
4894:
3427:
2325:, as shown by the following proposition:
2118:{\displaystyle Y=\{0\}\times \mathbb {R} }
226:of the Minkowski sum of the complement of
5123:The Minkowski Sum of a Disk and a Polygon
5114:Computational Geometry Algorithms Library
4999:Convex bodies: the Brunn-Minkowski theory
4996:
4987:
4967:
4950:
4844:
4792:
4774:
4671:
4606:Convex bodies: The Brunn–Minkowski theory
4602:
4570:Convex bodies: The Brunn–Minkowski theory
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4000:
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2773:An example of a non-convex set such that
2259:
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2022:
1741:
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1447:
1400:
1375:
1347:
785:
665:
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270:
264:
214:produces a set that could be summed with
121:
115:
5913:Group algebra of a locally compact group
5048:European Journal of Operational Research
5041:
5023:European Journal of Operational Research
4732:European Journal of Operational Research
4701:European Journal of Operational Research
4694:
4376:
3313:
2768:
2285:consisting of everything other than the
1329:{\displaystyle S+\emptyset =\emptyset .}
973:{\displaystyle B=\{(0,0),(1,1),(1,-1)\}}
884:{\displaystyle A=\{(1,0),(0,1),(0,-1)\}}
742:
29:
5272:Locally convex topological vector space
5001:, Cambridge: Cambridge University Press
4857:; Intriligator, Michael D (eds.).
4830:
3360:vertices and may be computed in time O(
3310:Algorithms for computing Minkowski sums
14:
6802:
6246:Uniform boundedness (Banach–Steinhaus)
4808: ed.). Amsterdam: North-Holland.
3289:
2582:of Minkowski summation and of forming
182:) is the corresponding inverse, where
6088:
5604:
5163:
5133:Minkowski's addition of convex shapes
5006:
4975:Discrete & Computational Geometry
4756:
4653:
4506:Theorem 3 (pages 562–563):
4449:Lozano-Pérez, Tomás (February 1983).
3388:. Imagine that these edges are solid
3281:
5091:
4082:{\displaystyle h_{K+L}=h_{K}+h_{L}.}
3269:
2645:is also a convex set; furthermore
2180:then the Minkowski sum of these two
5007:Tao, Terence & Vu, Van (2006),
4348:Topological vector space#Properties
3975:
2729:{\displaystyle \mu ,\lambda \geq 0}
2078:{\displaystyle f(x)={\frac {1}{x}}}
1964:other set will be an open subset.
1664:{\displaystyle B_{r}+B_{s}=B_{r+s}}
24:
5127:The Wolfram Demonstrations Project
5119:The Minkowski Sum of Two Triangles
4420:
3968:where "ess sup" denotes the
3877:
3874:
3871:
3867:
3864:
3861:
3834:
3563:
3516:
3237:
2173:{\displaystyle X=\mathbb {R} ^{2}}
1827:
1594:
1508:
1320:
1314:
455:{\displaystyle (A+B)-B\supseteq A}
410:{\displaystyle (A-B)+B\subseteq A}
25:
6866:
5067:
5044:"Aggregation of scale efficiency"
4697:"Aggregation of scale efficiency"
3106:{\displaystyle B+B\subsetneq 2B.}
2933:It can be easily calculated that
2845:{\displaystyle A+A\subsetneq 2A.}
2578:In mathematical terminology, the
2269:
1748:{\displaystyle c\in \mathbb {K} }
760:For example, if we have two sets
6784:
6783:
6069:
6068:
5995:Topological quantum field theory
5017:Mayer, A.; Zelenyuk, V. (2014).
4955:, GTM, vol. 165, Springer,
4726:Mayer, A.; Zelenyuk, V. (2014).
4147:{\displaystyle \mathbb {R} ^{n}}
4013:{\displaystyle \mathbb {R} ^{n}}
3329:Two convex polygons in the plane
3254:Numerical control (NC) machining
798:{\displaystyle \mathbb {R} ^{2}}
684:
667:
655:
647:
272:
260:
257:
140:
123:
111:
103:
6771:With the approximation property
5377:Ekeland's variational principle
4750:
4719:
3384:into a single ordered sequence
3210:(with various uses, notably by
3193:
3068:{\displaystyle B+B=\cup \cup ,}
2638:{\displaystyle \mu S+\lambda S}
1788:is defined (which happens when
1720:{\displaystyle cB_{r}=B_{|c|r}}
6234:Open mapping (Banach–Schauder)
4688:
4647:
4631:
4595:
4547:
4500:
4459:IEEE Transactions on Computers
4442:
4414:
4370:
3949:
3937:
3924:
3918:
3852:
3846:
3790:
3778:
3765:
3759:
3716:
3710:
3631:
3619:
3600:
3510:
3498:
3487:
3432:There is also a notion of the
3323:
3059:
3047:
3041:
3029:
3023:
3011:
2979:
2967:
2961:
2949:
2917:
2905:
2899:
2887:
2685:
2673:
2559:
2546:
2474:
2461:
2449:
2436:
2424:
2398:
2317:Convex hulls of Minkowski sums
2221:
2209:
2059:
2053:
2003:
1983:
1771:
1763:
1708:
1700:
1597:
1585:
1511:
1499:
1463:
1455:
1173:
1158:
1152:
1140:
1134:
1119:
1113:
1101:
1095:
1083:
1077:
1065:
1059:
1044:
1038:
1026:
1020:
1008:
964:
949:
943:
931:
925:
913:
875:
860:
854:
842:
836:
824:
715:
706:
661:
579:
575:
566:
550:
511:
507:
498:
482:
437:
425:
392:
380:
344:
340:
331:
315:
266:
201:
189:
117:
13:
1:
5791:Uniform boundedness principle
5145:by Marius Kintel: Application
4951:Nathanson, Melvyn B. (1996),
4870:10.1016/S1573-4382(81)01005-9
4846:10.1090/S0273-0979-02-00941-2
4833:Bull. Amer. Math. Soc. (N.S.)
4786:
4638:The Theorem of Barbier (Java)
2759:{\displaystyle \mu ,\lambda }
1833:{\displaystyle r\neq \infty }
1486:is the closed ball of radius
1410:{\displaystyle \mathbb {C} .}
1357:{\displaystyle \mathbb {K} ,}
5011:, Cambridge University Press
4918:Rockafellar, R. Tyrrell
4802:General competitive analysis
4660:Discrete Applied Mathematics
3179:{\displaystyle 180^{\circ }}
3139:, then the Minkowski sum of
1559:{\displaystyle \mathbb {K} }
1382:{\displaystyle \mathbb {R} }
1198:For Minkowski addition, the
983:then their Minkowski sum is
7:
6850:Theorems in convex geometry
6455:Radially convex/Star-shaped
6440:Pre-compact/Totally bounded
5397:Hermite–Hadamard inequality
5080:Encyclopedia of Mathematics
4839:(3): 355–405 (electronic),
4757:Firey, William J. (1962), "
4281: – theorem in geometry
4266:
3376:of the directed edges from
3374:merge the ordered sequences
3242:Minkowski sums are used in
2804:{\displaystyle A+A\neq 2A.}
2766:, then the set is convex.
10:
6871:
6141:Continuous linear operator
5934:Invariant subspace problem
5060:10.1016/j.ejor.2014.06.038
5035:10.1016/j.ejor.2014.04.003
4855:Arrow, Kenneth Joseph
4776:10.7146/math.scand.a-10510
4761:-means of convex bodies",
4744:10.1016/j.ejor.2014.04.003
4713:10.1016/j.ejor.2014.06.038
3991:compact convex subsets in
2329:For all non-empty subsets
1280:{\displaystyle S+\{0\}=S.}
738:
222:. This is defined as the
6779:
6524:
6486:Algebraic interior (core)
6468:
6366:
6254:
6228:Vector-valued Hahn–Banach
6189:
6123:
6116:Topological vector spaces
6064:
6023:
5947:
5926:
5885:
5824:
5766:
5712:
5654:
5647:
5582:
5549:
5504:
5435:
5361:
5285:
5227:
5201:
4989:10.1007/s00454-005-1206-y
4673:10.1016/j.dam.2019.02.042
4154:containing the origin as
3204:brush-and-stroke paradigm
1727:will hold for any scalar
731:The concept is named for
6316:Topological homomorphism
6176:Topological vector space
5903:Spectrum of a C*-algebra
5583:Applications and related
5387:Fenchel-Young inequality
4997:Schneider, Rolf (1993),
4603:Schneider, Rolf (1993).
4567:Schneider, Rolf (1993).
4363:
4263:Brunn-Minkowski theory.
4117:of compact convex sets
3411:
2985:{\displaystyle 2B=\cup }
2926:{\displaystyle B=\cup .}
1606:{\displaystyle r,s\in ,}
6000:Noncommutative geometry
5343:Legendre transformation
5267:Legendre transformation
4654:Kline, Jeffery (2019).
4472:10.1109/TC.1983.1676196
4377:Hadwiger, Hugo (1950),
4279:Brunn–Minkowski theorem
4096:≥ 1, Firey defined the
3542:essential Minkowski sum
3434:essential Minkowski sum
3428:Essential Minkowski sum
3200:mathematical morphology
3137:curve of constant width
3135:is (the interior of) a
2736:. Conversely, if this "
1807:{\displaystyle c\neq 0}
230:with the reflection of
176:Minkowski decomposition
6374:Absolutely convex/disk
6056:Tomita–Takesaki theory
6031:Approximation property
5975:Calculus of variations
5590:Convexity in economics
5524:(lower) ideally convex
5382:Fenchel–Moreau theorem
5372:Carathéodory's theorem
5009:Additive Combinatorics
4794:Arrow, Kenneth J.
4235:
4148:
4083:
4014:
3959:
3800:
3657:
3531:
3320:
3228:Earth mover's distance
3180:
3153:
3129:
3107:
3069:
2986:
2927:
2868:
2846:
2811:
2805:
2760:
2730:
2695:
2639:
2607:
2569:
2484:
2377:
2350:
2299:
2279:
2174:
2139:
2119:
2079:
2033:
1950:
1930:
1910:
1883:
1863:
1834:
1808:
1782:
1755:such that the product
1749:
1721:
1665:
1607:
1560:
1538:
1518:
1480:
1411:
1383:
1358:
1330:
1281:
1225:
1224:{\displaystyle \{0\},}
1186:
974:
885:
799:
757:
722:
592:
524:
456:
411:
357:
289:
208:
157:
35:
6409:Complemented subspace
6223:hyperplane separation
6051:Banach–Mazur distance
6014:Generalized functions
5512:Convex series related
5412:Shapley–Folkman lemma
4513:Annals of Mathematics
4421:Li, Wei (Fall 2011).
4333:Shapley–Folkman lemma
4236:
4149:
4084:
4015:
3960:
3801:
3658:
3532:
3317:
3181:
3154:
3130:
3108:
3070:
2987:
2928:
2869:
2847:
2806:
2772:
2761:
2738:distributive property
2731:
2696:
2640:
2613:is a convex set then
2608:
2570:
2485:
2378:
2376:{\displaystyle S_{2}}
2351:
2349:{\displaystyle S_{1}}
2300:
2280:
2175:
2140:
2120:
2080:
2034:
1951:
1931:
1911:
1909:{\displaystyle B_{r}}
1884:
1864:
1835:
1809:
1783:
1750:
1722:
1666:
1608:
1561:
1539:
1519:
1517:{\displaystyle r\in }
1481:
1412:
1384:
1359:
1331:
1282:
1226:
1187:
975:
886:
800:
746:
723:
593:
525:
457:
412:
358:
290:
209:
207:{\displaystyle (A-B)}
172:Minkowski subtraction
158:
33:
6855:Variational analysis
6835:Geometric algorithms
6810:Abelian group theory
6659:Locally convex space
6209:Closed graph theorem
6161:Locally convex space
5796:Kakutani fixed-point
5781:Riesz representation
5402:Krein–Milman theorem
5195:variational analysis
5075:"Minkowski addition"
5042:Zelenyuk, V (2015).
4695:Zelenyuk, V (2015).
4246:Minkowski inequality
4161:
4129:
4031:
4024:of the convex sets:
3995:
3816:
3809:it can be seen that
3689:
3551:
3447:
3302:for convex hulls in
3230:, and by extension,
3224:3D computer graphics
3208:2D computer graphics
3163:
3143:
3119:
3079:
2996:
2937:
2878:
2858:
2818:
2777:
2744:
2708:
2652:
2617:
2597:
2500:
2389:
2360:
2333:
2289:
2191:
2184:of the plane is the
2149:
2129:
2089:
2047:
1971:
1940:
1920:
1893:
1873:
1862:{\displaystyle r,s,}
1844:
1818:
1792:
1781:{\displaystyle |c|r}
1759:
1731:
1675:
1616:
1570:
1548:
1528:
1490:
1421:
1396:
1371:
1364:which is either the
1343:
1305:
1250:
1231:containing only the
1206:
990:
901:
812:
780:
628:
535:
467:
422:
377:
300:
241:
186:
180:geometric difference
168:Minkowski difference
84:
6639:Interpolation space
6171:Operator topologies
5980:Functional calculus
5939:Mahler's conjecture
5918:Von Neumann algebra
5632:Functional analysis
5392:Jensen's inequality
5262:Lagrange multiplier
5252:Convex optimization
5247:Convex metric space
5137:Alexander Bogomolny
4798:Hahn, Frank H.
4307:Interval arithmetic
4227:
4209:
4191:
3680:indicator functions
3300:collision detection
3290:Collision detection
3248:configuration space
3202:. It arises in the
1243:of a vector space,
1239:: for every subset
805:, with coordinates
6669:(Pseudo)Metrizable
6501:Minkowski addition
6353:Sublinear function
6005:Riemann hypothesis
5704:Topological vector
5520:(cs, bcs)-complete
5491:Algebraic interior
5209:Convex combination
4395:10.1007/BF01175656
4231:
4213:
4195:
4164:
4144:
4079:
4010:
3970:essential supremum
3955:
3904:
3796:
3748:
3653:
3527:
3344:in the plane with
3321:
3282:Aggregation theory
3176:
3149:
3125:
3103:
3065:
2982:
2923:
2864:
2842:
2812:
2801:
2756:
2726:
2691:
2635:
2603:
2565:
2480:
2373:
2346:
2295:
2275:
2170:
2135:
2115:
2075:
2029:
1946:
1926:
1906:
1879:
1859:
1830:
1804:
1778:
1745:
1717:
1661:
1603:
1556:
1534:
1514:
1476:
1407:
1379:
1354:
1326:
1277:
1221:
1182:
970:
881:
795:
758:
718:
619:vector subtraction
588:
520:
452:
407:
353:
285:
234:about the origin.
204:
153:
73:to each vector in
67:adding each vector
36:
6840:Hermann Minkowski
6820:Binary operations
6797:
6796:
6516:Relative interior
6262:Bilinear operator
6146:Linear functional
6082:
6081:
5985:Integral operator
5762:
5761:
5598:
5597:
5105:, Yale University
4935:978-0-691-01586-6
4909:978-0-88275-418-5
4879:978-0-444-86126-9
4815:978-0-444-85497-1
4616:978-0-521-35220-8
4580:978-0-521-35220-8
4515:. Second Series.
3858:
3722:
3270:3D solid modeling
3260:numerical control
3232:optimal transport
3188:Barbier's theorem
3152:{\displaystyle K}
3128:{\displaystyle K}
2867:{\displaystyle 1}
2606:{\displaystyle S}
2298:{\displaystyle y}
2138:{\displaystyle y}
2073:
1949:{\displaystyle 0}
1929:{\displaystyle 0}
1882:{\displaystyle c}
1537:{\displaystyle 0}
733:Hermann Minkowski
682:
138:
16:(Redirected from
6862:
6830:Digital geometry
6787:
6786:
6761:Uniformly smooth
6430:
6422:
6389:Balanced/Circled
6379:Absorbing/Radial
6109:
6102:
6095:
6086:
6085:
6072:
6071:
5990:Jones polynomial
5908:Operator algebra
5652:
5651:
5625:
5618:
5611:
5602:
5601:
5516:(cs, lcs)-closed
5462:Effective domain
5417:Robinson–Ursescu
5293:Convex conjugate
5184:
5177:
5170:
5161:
5160:
5125:by George Beck,
5106:
5088:
5063:
5038:
5012:
5002:
4992:
4991:
4963:
4947:
4913:
4891:
4849:
4848:
4827:
4780:
4779:
4778:
4754:
4748:
4747:
4723:
4717:
4716:
4692:
4686:
4685:
4675:
4651:
4645:
4635:
4629:
4628:
4599:
4593:
4592:
4551:
4545:
4544:
4504:
4498:
4497:
4495:
4494:
4455:
4446:
4440:
4439:
4437:
4436:
4431:. pp. 13–14
4418:
4412:
4411:
4410:
4409:
4374:
4359:
4344:
4321:intrinsic volume
4317:Quermassintegral
4284:
4240:
4238:
4237:
4232:
4226:
4221:
4208:
4203:
4190:
4185:
4181:
4180:
4153:
4151:
4150:
4145:
4143:
4142:
4137:
4116:
4088:
4086:
4085:
4080:
4075:
4074:
4062:
4061:
4049:
4048:
4022:support function
4019:
4017:
4016:
4011:
4009:
4008:
4003:
3964:
3962:
3961:
3956:
3936:
3935:
3917:
3916:
3903:
3902:
3901:
3896:
3881:
3880:
3845:
3844:
3839:
3838:
3837:
3805:
3803:
3802:
3797:
3777:
3776:
3758:
3757:
3747:
3746:
3745:
3740:
3709:
3708:
3676:Lebesgue measure
3662:
3660:
3659:
3654:
3649:
3645:
3638:
3634:
3603:
3597:
3596:
3591:
3568:
3567:
3566:
3536:
3534:
3533:
3528:
3523:
3519:
3490:
3484:
3483:
3478:
3185:
3183:
3182:
3177:
3175:
3174:
3158:
3156:
3155:
3150:
3134:
3132:
3131:
3126:
3112:
3110:
3109:
3104:
3074:
3072:
3071:
3066:
2991:
2989:
2988:
2983:
2932:
2930:
2929:
2924:
2873:
2871:
2870:
2865:
2851:
2849:
2848:
2843:
2810:
2808:
2807:
2802:
2765:
2763:
2762:
2757:
2735:
2733:
2732:
2727:
2700:
2698:
2697:
2692:
2644:
2642:
2641:
2636:
2612:
2610:
2609:
2604:
2574:
2572:
2571:
2566:
2558:
2557:
2533:
2529:
2528:
2527:
2526:
2489:
2487:
2486:
2481:
2473:
2472:
2448:
2447:
2423:
2422:
2410:
2409:
2382:
2380:
2379:
2374:
2372:
2371:
2355:
2353:
2352:
2347:
2345:
2344:
2304:
2302:
2301:
2296:
2284:
2282:
2281:
2276:
2268:
2267:
2262:
2238:
2237:
2232:
2179:
2177:
2176:
2171:
2169:
2168:
2163:
2144:
2142:
2141:
2136:
2124:
2122:
2121:
2116:
2114:
2084:
2082:
2081:
2076:
2074:
2066:
2038:
2036:
2035:
2030:
2025:
1999:
1955:
1953:
1952:
1947:
1935:
1933:
1932:
1927:
1915:
1913:
1912:
1907:
1905:
1904:
1888:
1886:
1885:
1880:
1868:
1866:
1865:
1860:
1839:
1837:
1836:
1831:
1813:
1811:
1810:
1805:
1787:
1785:
1784:
1779:
1774:
1766:
1754:
1752:
1751:
1746:
1744:
1726:
1724:
1723:
1718:
1716:
1715:
1711:
1703:
1690:
1689:
1670:
1668:
1667:
1662:
1660:
1659:
1641:
1640:
1628:
1627:
1612:
1610:
1609:
1604:
1565:
1563:
1562:
1557:
1555:
1543:
1541:
1540:
1535:
1523:
1521:
1520:
1515:
1485:
1483:
1482:
1477:
1466:
1458:
1450:
1433:
1432:
1416:
1414:
1413:
1408:
1403:
1388:
1386:
1385:
1380:
1378:
1363:
1361:
1360:
1355:
1350:
1335:
1333:
1332:
1327:
1286:
1284:
1283:
1278:
1237:identity element
1230:
1228:
1227:
1222:
1191:
1189:
1188:
1183:
979:
977:
976:
971:
890:
888:
887:
882:
804:
802:
801:
796:
794:
793:
788:
756:
727:
725:
724:
719:
687:
680:
670:
664:
658:
650:
603:image processing
597:
595:
594:
589:
587:
586:
562:
561:
529:
527:
526:
521:
519:
518:
494:
493:
461:
459:
458:
453:
416:
414:
413:
408:
362:
360:
359:
354:
352:
351:
327:
326:
294:
292:
291:
286:
275:
269:
263:
213:
211:
210:
205:
162:
160:
159:
154:
143:
136:
126:
120:
114:
106:
52:position vectors
21:
6870:
6869:
6865:
6864:
6863:
6861:
6860:
6859:
6825:Convex geometry
6815:Affine geometry
6800:
6799:
6798:
6793:
6775:
6537:B-complete/Ptak
6520:
6464:
6428:
6420:
6399:Bounding points
6362:
6304:Densely defined
6250:
6239:Bounded inverse
6185:
6119:
6113:
6083:
6078:
6060:
6024:Advanced topics
6019:
5943:
5922:
5881:
5847:Hilbert–Schmidt
5820:
5811:Gelfand–Naimark
5758:
5708:
5643:
5629:
5599:
5594:
5578:
5545:
5500:
5431:
5357:
5348:Semi-continuity
5333:Convex function
5314:Logarithmically
5281:
5242:Convex geometry
5223:
5214:Convex function
5197:
5191:Convex analysis
5188:
5073:
5070:
4936:
4922:Convex analysis
4910:
4880:
4816:
4789:
4784:
4783:
4755:
4751:
4724:
4720:
4693:
4689:
4652:
4648:
4636:
4632:
4617:
4600:
4596:
4581:
4555:convexification
4552:
4548:
4525:10.2307/1968735
4505:
4501:
4492:
4490:
4453:
4447:
4443:
4434:
4432:
4419:
4415:
4407:
4405:
4375:
4371:
4366:
4357:
4342:
4282:
4269:
4257:
4255:
4248:, the function
4222:
4217:
4204:
4199:
4186:
4176:
4172:
4168:
4162:
4159:
4158:
4138:
4133:
4132:
4130:
4127:
4126:
4112:
4103:
4070:
4066:
4057:
4053:
4038:
4034:
4032:
4029:
4028:
4004:
3999:
3998:
3996:
3993:
3992:
3981:
3931:
3927:
3912:
3908:
3897:
3892:
3891:
3882:
3860:
3859:
3833:
3832:
3828:
3823:
3819:
3817:
3814:
3813:
3772:
3768:
3753:
3749:
3741:
3736:
3735:
3726:
3696:
3692:
3690:
3687:
3686:
3612:
3608:
3599:
3592:
3587:
3586:
3579:
3575:
3562:
3561:
3557:
3552:
3549:
3548:
3486:
3479:
3474:
3473:
3466:
3462:
3448:
3445:
3444:
3439:
3430:
3414:
3407:
3403:
3398:polygonal chain
3395:
3387:
3383:
3379:
3367:
3363:
3359:
3355:
3351:
3347:
3343:
3339:
3335:convex polygons
3331:
3326:
3312:
3304:physics engines
3292:
3284:
3272:
3256:
3244:motion planning
3240:
3238:Motion planning
3212:Donald E. Knuth
3196:
3170:
3166:
3164:
3161:
3160:
3144:
3141:
3140:
3120:
3117:
3116:
3080:
3077:
3076:
2997:
2994:
2993:
2938:
2935:
2934:
2879:
2876:
2875:
2859:
2856:
2855:
2819:
2816:
2815:
2778:
2775:
2774:
2745:
2742:
2741:
2709:
2706:
2705:
2653:
2650:
2649:
2618:
2615:
2614:
2598:
2595:
2594:
2553:
2549:
2522:
2518:
2517:
2513:
2509:
2501:
2498:
2497:
2468:
2464:
2443:
2439:
2418:
2414:
2405:
2401:
2390:
2387:
2386:
2367:
2363:
2361:
2358:
2357:
2340:
2336:
2334:
2331:
2330:
2319:
2290:
2287:
2286:
2263:
2258:
2257:
2233:
2228:
2227:
2192:
2189:
2188:
2164:
2159:
2158:
2150:
2147:
2146:
2130:
2127:
2126:
2110:
2090:
2087:
2086:
2065:
2048:
2045:
2044:
2021:
1995:
1972:
1969:
1968:
1941:
1938:
1937:
1921:
1918:
1917:
1900:
1896:
1894:
1891:
1890:
1874:
1871:
1870:
1845:
1842:
1841:
1819:
1816:
1815:
1793:
1790:
1789:
1770:
1762:
1760:
1757:
1756:
1740:
1732:
1729:
1728:
1707:
1699:
1698:
1694:
1685:
1681:
1676:
1673:
1672:
1649:
1645:
1636:
1632:
1623:
1619:
1617:
1614:
1613:
1571:
1568:
1567:
1551:
1549:
1546:
1545:
1529:
1526:
1525:
1491:
1488:
1487:
1462:
1454:
1446:
1428:
1424:
1422:
1419:
1418:
1399:
1397:
1394:
1393:
1391:complex numbers
1374:
1372:
1369:
1368:
1346:
1344:
1341:
1340:
1306:
1303:
1302:
1251:
1248:
1247:
1207:
1204:
1203:
991:
988:
987:
902:
899:
898:
813:
810:
809:
789:
784:
783:
781:
778:
777:
748:
741:
683:
666:
660:
654:
646:
629:
626:
625:
582:
578:
557:
553:
536:
533:
532:
514:
510:
489:
485:
468:
465:
464:
423:
420:
419:
378:
375:
374:
347:
343:
322:
318:
301:
298:
297:
271:
265:
256:
242:
239:
238:
187:
184:
183:
139:
122:
116:
110:
102:
85:
82:
81:
63:Euclidean space
28:
23:
22:
15:
12:
11:
5:
6868:
6858:
6857:
6852:
6847:
6842:
6837:
6832:
6827:
6822:
6817:
6812:
6795:
6794:
6792:
6791:
6780:
6777:
6776:
6774:
6773:
6768:
6763:
6758:
6756:Ultrabarrelled
6748:
6742:
6737:
6731:
6726:
6721:
6716:
6711:
6706:
6697:
6691:
6686:
6684:Quasi-complete
6681:
6679:Quasibarrelled
6676:
6671:
6666:
6661:
6656:
6651:
6646:
6641:
6636:
6631:
6626:
6621:
6620:
6619:
6609:
6604:
6599:
6594:
6589:
6584:
6579:
6574:
6569:
6559:
6554:
6544:
6539:
6534:
6528:
6526:
6522:
6521:
6519:
6518:
6508:
6503:
6498:
6493:
6488:
6478:
6472:
6470:
6469:Set operations
6466:
6465:
6463:
6462:
6457:
6452:
6447:
6442:
6437:
6432:
6424:
6416:
6411:
6406:
6401:
6396:
6391:
6386:
6381:
6376:
6370:
6368:
6364:
6363:
6361:
6360:
6355:
6350:
6345:
6340:
6339:
6338:
6333:
6328:
6318:
6313:
6312:
6311:
6306:
6301:
6296:
6291:
6286:
6281:
6271:
6270:
6269:
6258:
6256:
6252:
6251:
6249:
6248:
6243:
6242:
6241:
6231:
6225:
6216:
6211:
6206:
6204:Banach–Alaoglu
6201:
6199:Anderson–Kadec
6195:
6193:
6187:
6186:
6184:
6183:
6178:
6173:
6168:
6163:
6158:
6153:
6148:
6143:
6138:
6133:
6127:
6125:
6124:Basic concepts
6121:
6120:
6112:
6111:
6104:
6097:
6089:
6080:
6079:
6077:
6076:
6065:
6062:
6061:
6059:
6058:
6053:
6048:
6043:
6041:Choquet theory
6038:
6033:
6027:
6025:
6021:
6020:
6018:
6017:
6007:
6002:
5997:
5992:
5987:
5982:
5977:
5972:
5967:
5962:
5957:
5951:
5949:
5945:
5944:
5942:
5941:
5936:
5930:
5928:
5924:
5923:
5921:
5920:
5915:
5910:
5905:
5900:
5895:
5893:Banach algebra
5889:
5887:
5883:
5882:
5880:
5879:
5874:
5869:
5864:
5859:
5854:
5849:
5844:
5839:
5834:
5828:
5826:
5822:
5821:
5819:
5818:
5816:Banach–Alaoglu
5813:
5808:
5803:
5798:
5793:
5788:
5783:
5778:
5772:
5770:
5764:
5763:
5760:
5759:
5757:
5756:
5751:
5746:
5744:Locally convex
5741:
5727:
5722:
5716:
5714:
5710:
5709:
5707:
5706:
5701:
5696:
5691:
5686:
5681:
5676:
5671:
5666:
5661:
5655:
5649:
5645:
5644:
5628:
5627:
5620:
5613:
5605:
5596:
5595:
5593:
5592:
5586:
5584:
5580:
5579:
5577:
5576:
5571:
5569:Strong duality
5566:
5561:
5555:
5553:
5547:
5546:
5544:
5543:
5508:
5506:
5502:
5501:
5499:
5498:
5493:
5484:
5479:
5477:John ellipsoid
5474:
5469:
5464:
5459:
5445:
5439:
5437:
5433:
5432:
5430:
5429:
5424:
5419:
5414:
5409:
5404:
5399:
5394:
5389:
5384:
5379:
5374:
5368:
5366:
5364:results (list)
5359:
5358:
5356:
5355:
5350:
5345:
5340:
5338:Invex function
5335:
5326:
5321:
5316:
5311:
5306:
5300:
5295:
5289:
5287:
5283:
5282:
5280:
5279:
5274:
5269:
5264:
5259:
5254:
5249:
5244:
5239:
5237:Choquet theory
5233:
5231:
5225:
5224:
5222:
5221:
5216:
5211:
5205:
5203:
5202:Basic concepts
5199:
5198:
5187:
5186:
5179:
5172:
5164:
5158:
5157:
5152:
5151:by Joan Gerard
5146:
5140:
5130:
5116:
5110:Minkowski Sums
5107:
5089:
5069:
5068:External links
5066:
5065:
5064:
5054:(1): 269–277.
5039:
5029:(3): 774–785.
5014:
5004:
4994:
4982:(2): 223–240,
4965:
4948:
4934:
4914:
4908:
4892:
4878:
4850:
4828:
4814:
4788:
4785:
4782:
4781:
4749:
4738:(3): 774–785.
4718:
4707:(1): 269–277.
4687:
4646:
4630:
4615:
4594:
4579:
4546:
4519:(3): 556–583.
4499:
4441:
4413:
4389:(3): 210–218,
4368:
4367:
4365:
4362:
4361:
4360:
4351:
4345:
4336:
4330:
4327:Parallel curve
4324:
4310:
4304:
4298:
4292:
4286:
4276:
4268:
4265:
4253:
4251:
4242:
4241:
4230:
4225:
4220:
4216:
4212:
4207:
4202:
4198:
4194:
4189:
4184:
4179:
4175:
4171:
4167:
4141:
4136:
4108:
4090:
4089:
4078:
4073:
4069:
4065:
4060:
4056:
4052:
4047:
4044:
4041:
4037:
4007:
4002:
3980:
3974:
3966:
3965:
3954:
3951:
3948:
3945:
3942:
3939:
3934:
3930:
3926:
3923:
3920:
3915:
3911:
3907:
3900:
3895:
3889:
3885:
3879:
3876:
3873:
3869:
3866:
3863:
3857:
3854:
3851:
3848:
3843:
3836:
3831:
3826:
3822:
3807:
3806:
3795:
3792:
3789:
3786:
3783:
3780:
3775:
3771:
3767:
3764:
3761:
3756:
3752:
3744:
3739:
3733:
3729:
3725:
3721:
3718:
3715:
3712:
3707:
3703:
3699:
3695:
3664:
3663:
3652:
3648:
3644:
3641:
3637:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3611:
3607:
3602:
3595:
3590:
3585:
3582:
3578:
3574:
3571:
3565:
3560:
3556:
3544:is defined by
3538:
3537:
3526:
3522:
3518:
3515:
3512:
3509:
3506:
3503:
3500:
3497:
3494:
3489:
3482:
3477:
3472:
3469:
3465:
3461:
3458:
3455:
3452:
3437:
3429:
3426:
3413:
3410:
3405:
3401:
3393:
3385:
3381:
3377:
3365:
3361:
3357:
3353:
3349:
3345:
3341:
3337:
3330:
3327:
3325:
3322:
3311:
3308:
3296:GJK algorithms
3291:
3288:
3283:
3280:
3271:
3268:
3255:
3252:
3239:
3236:
3218:), and as the
3195:
3192:
3173:
3169:
3148:
3124:
3102:
3099:
3096:
3093:
3090:
3087:
3084:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2922:
2919:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2874:dimension is:
2863:
2854:An example in
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2755:
2752:
2749:
2725:
2722:
2719:
2716:
2713:
2702:
2701:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2634:
2631:
2628:
2625:
2622:
2602:
2576:
2575:
2564:
2561:
2556:
2552:
2548:
2545:
2542:
2539:
2536:
2532:
2525:
2521:
2516:
2512:
2508:
2505:
2491:
2490:
2479:
2476:
2471:
2467:
2463:
2460:
2457:
2454:
2451:
2446:
2442:
2438:
2435:
2432:
2429:
2426:
2421:
2417:
2413:
2408:
2404:
2400:
2397:
2394:
2384:
2370:
2366:
2343:
2339:
2318:
2315:
2311:compact subset
2294:
2274:
2271:
2266:
2261:
2256:
2253:
2250:
2247:
2244:
2241:
2236:
2231:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2182:closed subsets
2167:
2162:
2157:
2154:
2134:
2113:
2109:
2106:
2103:
2100:
2097:
2094:
2072:
2069:
2064:
2061:
2058:
2055:
2052:
2028:
2024:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1998:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1963:
1945:
1925:
1903:
1899:
1878:
1858:
1855:
1852:
1849:
1829:
1826:
1823:
1803:
1800:
1797:
1777:
1773:
1769:
1765:
1743:
1739:
1736:
1714:
1710:
1706:
1702:
1697:
1693:
1688:
1684:
1680:
1658:
1655:
1652:
1648:
1644:
1639:
1635:
1631:
1626:
1622:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1554:
1533:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1475:
1472:
1469:
1465:
1461:
1457:
1453:
1449:
1445:
1442:
1439:
1436:
1431:
1427:
1406:
1402:
1377:
1353:
1349:
1337:
1336:
1325:
1322:
1319:
1316:
1313:
1310:
1288:
1287:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1220:
1217:
1214:
1211:
1201:
1193:
1192:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
981:
980:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
892:
891:
880:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
792:
787:
747:Minkowski sum
740:
737:
729:
728:
717:
714:
711:
708:
705:
702:
699:
696:
693:
690:
686:
679:
676:
673:
669:
663:
657:
653:
649:
645:
642:
639:
636:
633:
599:
598:
585:
581:
577:
574:
571:
568:
565:
560:
556:
552:
549:
546:
543:
540:
530:
517:
513:
509:
506:
503:
500:
497:
492:
488:
484:
481:
478:
475:
472:
462:
451:
448:
445:
442:
439:
436:
433:
430:
427:
417:
406:
403:
400:
397:
394:
391:
388:
385:
382:
364:
363:
350:
346:
342:
339:
336:
333:
330:
325:
321:
317:
314:
311:
308:
305:
295:
284:
281:
278:
274:
268:
262:
259:
255:
252:
249:
246:
203:
200:
197:
194:
191:
164:
163:
152:
149:
146:
142:
135:
132:
129:
125:
119:
113:
109:
105:
101:
98:
95:
92:
89:
26:
9:
6:
4:
3:
2:
6867:
6856:
6853:
6851:
6848:
6846:
6843:
6841:
6838:
6836:
6833:
6831:
6828:
6826:
6823:
6821:
6818:
6816:
6813:
6811:
6808:
6807:
6805:
6790:
6782:
6781:
6778:
6772:
6769:
6767:
6764:
6762:
6759:
6757:
6753:
6749:
6747:) convex
6746:
6743:
6741:
6738:
6736:
6732:
6730:
6727:
6725:
6722:
6720:
6719:Semi-complete
6717:
6715:
6712:
6710:
6707:
6705:
6701:
6698:
6696:
6692:
6690:
6687:
6685:
6682:
6680:
6677:
6675:
6672:
6670:
6667:
6665:
6662:
6660:
6657:
6655:
6652:
6650:
6647:
6645:
6642:
6640:
6637:
6635:
6634:Infrabarreled
6632:
6630:
6627:
6625:
6622:
6618:
6615:
6614:
6613:
6610:
6608:
6605:
6603:
6600:
6598:
6595:
6593:
6592:Distinguished
6590:
6588:
6585:
6583:
6580:
6578:
6575:
6573:
6570:
6568:
6564:
6560:
6558:
6555:
6553:
6549:
6545:
6543:
6540:
6538:
6535:
6533:
6530:
6529:
6527:
6525:Types of TVSs
6523:
6517:
6513:
6509:
6507:
6504:
6502:
6499:
6497:
6494:
6492:
6489:
6487:
6483:
6479:
6477:
6474:
6473:
6471:
6467:
6461:
6458:
6456:
6453:
6451:
6448:
6446:
6445:Prevalent/Shy
6443:
6441:
6438:
6436:
6435:Extreme point
6433:
6431:
6425:
6423:
6417:
6415:
6412:
6410:
6407:
6405:
6402:
6400:
6397:
6395:
6392:
6390:
6387:
6385:
6382:
6380:
6377:
6375:
6372:
6371:
6369:
6367:Types of sets
6365:
6359:
6356:
6354:
6351:
6349:
6346:
6344:
6341:
6337:
6334:
6332:
6329:
6327:
6324:
6323:
6322:
6319:
6317:
6314:
6310:
6309:Discontinuous
6307:
6305:
6302:
6300:
6297:
6295:
6292:
6290:
6287:
6285:
6282:
6280:
6277:
6276:
6275:
6272:
6268:
6265:
6264:
6263:
6260:
6259:
6257:
6253:
6247:
6244:
6240:
6237:
6236:
6235:
6232:
6229:
6226:
6224:
6220:
6217:
6215:
6212:
6210:
6207:
6205:
6202:
6200:
6197:
6196:
6194:
6192:
6188:
6182:
6179:
6177:
6174:
6172:
6169:
6167:
6166:Metrizability
6164:
6162:
6159:
6157:
6154:
6152:
6151:Fréchet space
6149:
6147:
6144:
6142:
6139:
6137:
6134:
6132:
6129:
6128:
6126:
6122:
6117:
6110:
6105:
6103:
6098:
6096:
6091:
6090:
6087:
6075:
6067:
6066:
6063:
6057:
6054:
6052:
6049:
6047:
6046:Weak topology
6044:
6042:
6039:
6037:
6034:
6032:
6029:
6028:
6026:
6022:
6015:
6011:
6008:
6006:
6003:
6001:
5998:
5996:
5993:
5991:
5988:
5986:
5983:
5981:
5978:
5976:
5973:
5971:
5970:Index theorem
5968:
5966:
5963:
5961:
5958:
5956:
5953:
5952:
5950:
5946:
5940:
5937:
5935:
5932:
5931:
5929:
5927:Open problems
5925:
5919:
5916:
5914:
5911:
5909:
5906:
5904:
5901:
5899:
5896:
5894:
5891:
5890:
5888:
5884:
5878:
5875:
5873:
5870:
5868:
5865:
5863:
5860:
5858:
5855:
5853:
5850:
5848:
5845:
5843:
5840:
5838:
5835:
5833:
5830:
5829:
5827:
5823:
5817:
5814:
5812:
5809:
5807:
5804:
5802:
5799:
5797:
5794:
5792:
5789:
5787:
5784:
5782:
5779:
5777:
5774:
5773:
5771:
5769:
5765:
5755:
5752:
5750:
5747:
5745:
5742:
5739:
5735:
5731:
5728:
5726:
5723:
5721:
5718:
5717:
5715:
5711:
5705:
5702:
5700:
5697:
5695:
5692:
5690:
5687:
5685:
5682:
5680:
5677:
5675:
5672:
5670:
5667:
5665:
5662:
5660:
5657:
5656:
5653:
5650:
5646:
5641:
5637:
5633:
5626:
5621:
5619:
5614:
5612:
5607:
5606:
5603:
5591:
5588:
5587:
5585:
5581:
5575:
5572:
5570:
5567:
5565:
5562:
5560:
5557:
5556:
5554:
5552:
5548:
5541:
5539:
5533:
5531:
5525:
5521:
5517:
5513:
5510:
5509:
5507:
5503:
5497:
5494:
5492:
5488:
5485:
5483:
5480:
5478:
5475:
5473:
5470:
5468:
5465:
5463:
5460:
5458:
5454:
5450:
5446:
5444:
5441:
5440:
5438:
5434:
5428:
5425:
5423:
5420:
5418:
5415:
5413:
5410:
5408:
5407:Mazur's lemma
5405:
5403:
5400:
5398:
5395:
5393:
5390:
5388:
5385:
5383:
5380:
5378:
5375:
5373:
5370:
5369:
5367:
5365:
5360:
5354:
5353:Subderivative
5351:
5349:
5346:
5344:
5341:
5339:
5336:
5334:
5330:
5327:
5325:
5322:
5320:
5317:
5315:
5312:
5310:
5307:
5305:
5301:
5299:
5296:
5294:
5291:
5290:
5288:
5284:
5278:
5275:
5273:
5270:
5268:
5265:
5263:
5260:
5258:
5255:
5253:
5250:
5248:
5245:
5243:
5240:
5238:
5235:
5234:
5232:
5230:
5229:Topics (list)
5226:
5220:
5217:
5215:
5212:
5210:
5207:
5206:
5204:
5200:
5196:
5192:
5185:
5180:
5178:
5173:
5171:
5166:
5165:
5162:
5156:
5153:
5150:
5147:
5144:
5141:
5138:
5134:
5131:
5128:
5124:
5120:
5117:
5115:
5111:
5108:
5104:
5100:
5099:
5094:
5090:
5086:
5082:
5081:
5076:
5072:
5071:
5061:
5057:
5053:
5049:
5045:
5040:
5036:
5032:
5028:
5024:
5020:
5015:
5010:
5005:
5000:
4995:
4990:
4985:
4981:
4977:
4976:
4971:
4970:Sharir, Micha
4968:Oks, Eduard;
4966:
4962:
4958:
4954:
4949:
4945:
4941:
4937:
4931:
4927:
4923:
4919:
4915:
4911:
4905:
4901:
4897:
4893:
4889:
4885:
4881:
4875:
4871:
4867:
4863:
4862:
4856:
4851:
4847:
4842:
4838:
4834:
4829:
4825:
4821:
4817:
4811:
4807:
4803:
4799:
4795:
4791:
4790:
4777:
4772:
4768:
4764:
4760:
4753:
4745:
4741:
4737:
4733:
4729:
4722:
4714:
4710:
4706:
4702:
4698:
4691:
4683:
4679:
4674:
4669:
4665:
4661:
4657:
4650:
4643:
4639:
4634:
4626:
4622:
4618:
4612:
4608:
4607:
4598:
4590:
4586:
4582:
4576:
4572:
4571:
4564:
4561:of Minkowski
4560:
4556:
4550:
4542:
4538:
4534:
4530:
4526:
4522:
4518:
4514:
4510:
4503:
4489:
4485:
4481:
4477:
4473:
4469:
4465:
4461:
4460:
4452:
4445:
4430:
4426:
4425:
4417:
4404:
4400:
4396:
4392:
4388:
4384:
4380:
4373:
4369:
4355:
4352:
4349:
4346:
4340:
4337:
4334:
4331:
4328:
4325:
4322:
4318:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4293:
4290:
4287:
4280:
4277:
4274:
4271:
4270:
4264:
4262:
4258:
4247:
4228:
4223:
4218:
4214:
4210:
4205:
4200:
4196:
4192:
4187:
4182:
4177:
4173:
4169:
4165:
4157:
4156:
4155:
4139:
4124:
4120:
4115:
4111:
4106:
4102:
4101:Minkowski sum
4100:
4095:
4076:
4071:
4067:
4063:
4058:
4054:
4050:
4045:
4042:
4039:
4035:
4027:
4026:
4025:
4023:
4005:
3990:
3986:
3979:Minkowski sum
3978:
3973:
3971:
3952:
3946:
3943:
3940:
3932:
3928:
3921:
3913:
3909:
3905:
3898:
3887:
3883:
3855:
3849:
3841:
3829:
3824:
3820:
3812:
3811:
3810:
3793:
3787:
3784:
3781:
3773:
3769:
3762:
3754:
3750:
3742:
3731:
3727:
3719:
3713:
3705:
3701:
3697:
3693:
3685:
3684:
3683:
3681:
3677:
3674:-dimensional
3673:
3669:
3650:
3646:
3642:
3639:
3635:
3628:
3625:
3622:
3616:
3613:
3609:
3605:
3593:
3583:
3580:
3576:
3572:
3569:
3558:
3554:
3547:
3546:
3545:
3543:
3524:
3520:
3513:
3507:
3504:
3501:
3495:
3492:
3480:
3470:
3467:
3463:
3459:
3456:
3453:
3450:
3443:
3442:
3441:
3435:
3425:
3423:
3419:
3409:
3399:
3391:
3375:
3371:
3336:
3316:
3307:
3305:
3301:
3297:
3287:
3279:
3277:
3267:
3265:
3264:cutting piece
3261:
3251:
3249:
3245:
3235:
3233:
3229:
3225:
3222:operation of
3221:
3217:
3213:
3209:
3205:
3201:
3191:
3189:
3171:
3167:
3146:
3138:
3122:
3113:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3062:
3056:
3053:
3050:
3044:
3038:
3035:
3032:
3026:
3020:
3017:
3014:
3008:
3005:
3002:
2999:
2976:
2973:
2970:
2964:
2958:
2955:
2952:
2946:
2943:
2940:
2920:
2914:
2911:
2908:
2902:
2896:
2893:
2890:
2884:
2881:
2861:
2852:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2771:
2767:
2753:
2750:
2747:
2739:
2723:
2720:
2717:
2714:
2711:
2688:
2682:
2679:
2676:
2670:
2667:
2664:
2661:
2658:
2655:
2648:
2647:
2646:
2632:
2629:
2626:
2623:
2620:
2600:
2591:
2589:
2585:
2581:
2562:
2554:
2550:
2543:
2540:
2537:
2534:
2530:
2523:
2519:
2514:
2510:
2506:
2503:
2496:
2495:
2494:
2477:
2469:
2465:
2458:
2455:
2452:
2444:
2440:
2433:
2430:
2427:
2419:
2415:
2411:
2406:
2402:
2395:
2392:
2385:
2368:
2364:
2341:
2337:
2328:
2327:
2326:
2324:
2314:
2312:
2308:
2292:
2272:
2264:
2254:
2248:
2245:
2242:
2239:
2234:
2224:
2218:
2215:
2212:
2203:
2200:
2197:
2194:
2187:
2183:
2165:
2155:
2152:
2132:
2107:
2101:
2095:
2092:
2070:
2067:
2062:
2056:
2050:
2042:
2018:
2015:
2012:
2009:
2006:
2000:
1996:
1992:
1989:
1986:
1977:
1974:
1965:
1961:
1959:
1943:
1923:
1901:
1897:
1876:
1856:
1853:
1850:
1847:
1824:
1821:
1801:
1798:
1795:
1775:
1767:
1737:
1734:
1712:
1704:
1695:
1691:
1686:
1682:
1678:
1656:
1653:
1650:
1646:
1642:
1637:
1633:
1629:
1624:
1620:
1600:
1591:
1588:
1582:
1579:
1576:
1573:
1566:then for any
1531:
1505:
1502:
1496:
1493:
1470:
1467:
1459:
1451:
1443:
1440:
1434:
1429:
1425:
1404:
1392:
1367:
1351:
1323:
1317:
1311:
1308:
1301:
1300:
1299:
1297:
1293:
1274:
1271:
1268:
1262:
1256:
1253:
1246:
1245:
1244:
1242:
1238:
1234:
1218:
1212:
1199:
1196:
1179:
1170:
1167:
1164:
1161:
1155:
1149:
1146:
1143:
1137:
1131:
1128:
1125:
1122:
1116:
1110:
1107:
1104:
1098:
1092:
1089:
1086:
1080:
1074:
1071:
1068:
1062:
1056:
1053:
1050:
1047:
1041:
1035:
1032:
1029:
1023:
1017:
1014:
1011:
1002:
999:
996:
993:
986:
985:
984:
961:
958:
955:
952:
946:
940:
937:
934:
928:
922:
919:
916:
907:
904:
897:
896:
895:
872:
869:
866:
863:
857:
851:
848:
845:
839:
833:
830:
827:
818:
815:
808:
807:
806:
790:
775:
771:
767:
763:
755:
751:
745:
736:
734:
712:
709:
703:
700:
697:
691:
688:
677:
674:
671:
651:
640:
637:
634:
631:
624:
623:
622:
620:
614:
612:
608:
604:
583:
572:
569:
563:
558:
554:
547:
544:
541:
538:
531:
515:
504:
501:
495:
490:
486:
479:
476:
473:
470:
463:
449:
446:
443:
440:
434:
431:
428:
418:
404:
401:
398:
395:
389:
386:
383:
373:
372:
371:
369:
348:
337:
334:
328:
323:
319:
312:
309:
306:
303:
296:
279:
276:
250:
247:
244:
237:
236:
235:
233:
229:
225:
221:
217:
198:
195:
192:
181:
177:
173:
169:
147:
144:
133:
130:
127:
107:
96:
93:
90:
87:
80:
79:
78:
76:
72:
68:
65:is formed by
64:
60:
56:
53:
49:
45:
44:Minkowski sum
41:
32:
19:
18:Minkowski sum
6695:Polynomially
6624:Grothendieck
6617:tame Fréchet
6567:Bornological
6500:
6427:Linear cone
6419:Convex cone
6394:Banach disks
6336:Sesquilinear
6191:Main results
6181:Vector space
6136:Completeness
6131:Banach space
6036:Balanced set
6010:Distribution
5948:Applications
5801:Krein–Milman
5786:Closed graph
5574:Weak duality
5537:
5529:
5449:Orthogonally
5097:
5078:
5051:
5047:
5026:
5022:
5008:
4998:
4979:
4973:
4952:
4925:
4921:
4899:
4860:
4858:
4836:
4832:
4805:
4801:
4766:
4763:Math. Scand.
4762:
4758:
4752:
4735:
4731:
4721:
4704:
4700:
4690:
4663:
4659:
4649:
4642:cut-the-knot
4633:
4605:
4597:
4569:
4559:convex hulls
4549:
4516:
4512:
4502:
4491:. Retrieved
4463:
4457:
4444:
4433:. Retrieved
4423:
4416:
4406:, retrieved
4386:
4382:
4372:
4313:Mixed volume
4273:Blaschke sum
4260:
4249:
4243:
4122:
4118:
4113:
4109:
4104:
4098:
4097:
4093:
4091:
3988:
3984:
3982:
3976:
3967:
3808:
3671:
3670:denotes the
3667:
3665:
3541:
3539:
3433:
3431:
3421:
3417:
3415:
3332:
3319:plus-signs.
3293:
3285:
3273:
3257:
3241:
3197:
3194:Applications
3114:
3075:hence again
2853:
2813:
2703:
2592:
2590:operations.
2584:convex hulls
2577:
2492:
2323:convex hulls
2320:
1966:
1524:centered at
1366:real numbers
1338:
1295:
1289:
1240:
1197:
1194:
982:
893:
765:
761:
759:
753:
749:
730:
615:
600:
367:
365:
231:
227:
219:
215:
179:
175:
171:
167:
165:
74:
70:
58:
54:
43:
37:
6689:Quasinormed
6602:FK-AK space
6496:Linear span
6491:Convex hull
6476:Affine hull
6279:Almost open
6219:Hahn–Banach
5965:Heat kernel
5955:Hardy space
5862:Trace class
5776:Hahn–Banach
5738:Topological
5564:Duality gap
5559:Dual system
5443:Convex hull
5139:: an applet
5093:Howe, Roger
4666:: 128–141.
4601:Chapter 1:
4480:1721.1/5684
4429:UC Berkeley
4289:Convolution
3370:polar angle
3324:Planar case
3298:to compute
3220:solid sweep
3159:and of its
2307:closed sets
2085:and if and
1958:open subset
1235:, 0, is an
1233:zero vector
218:to recover
6804:Categories
6729:Stereotype
6587:(DF)-space
6582:Convenient
6321:Functional
6289:Continuous
6274:Linear map
6214:F. Riesz's
6156:Linear map
5898:C*-algebra
5713:Properties
5487:Radial set
5457:Convex set
5219:Convex set
4961:0859.11003
4896:Henry Mann
4787:References
4493:2023-01-10
4466:(2): 111.
4435:2023-01-10
4408:2023-01-12
3540:Thus, the
2704:for every
2580:operations
224:complement
6745:Uniformly
6704:Reflexive
6552:Barrelled
6548:Countably
6460:Symmetric
6358:Transpose
5872:Unbounded
5867:Transpose
5825:Operators
5754:Separable
5749:Reflexive
5734:Algebraic
5720:Barrelled
5472:Hypograph
5085:EMS Press
4769:: 17–24,
4682:127962240
4509:Krein, M.
4403:121604732
3944:−
3906:
3888:∈
3785:−
3732:∈
3626:−
3617:∩
3606:μ
3584:∈
3517:∅
3514:≠
3505:−
3496:∩
3471:∈
3372:. Let us
3172:∘
3092:⊊
3045:∪
3027:∪
2965:∪
2903:∪
2831:⊊
2790:≠
2754:λ
2748:μ
2721:≥
2718:λ
2712:μ
2683:λ
2677:μ
2665:λ
2656:μ
2630:λ
2621:μ
2588:commuting
2544:
2538:∑
2515:∑
2507:
2459:
2434:
2396:
2270:∖
2246:≠
2225:∈
2145:-axis in
2108:×
2019:∈
2013:≠
1828:∞
1825:≠
1799:≠
1738:∈
1671:and also
1595:∞
1583:∈
1509:∞
1497:∈
1468:≤
1444:∈
1321:∅
1315:∅
1292:empty set
1168:−
1129:−
1054:−
959:−
870:−
774:triangles
710:−
689:∈
672:∈
652:−
635:−
584:∁
570:−
564:−
559:∁
516:∁
502:−
491:∁
474:−
447:⊇
441:−
402:⊆
387:−
349:∁
335:−
324:∁
307:−
277:∈
258:−
245:−
196:−
145:∈
128:∈
6789:Category
6740:Strictly
6714:Schwartz
6654:LF-space
6649:LB-space
6607:FK-space
6577:Complete
6557:BK-space
6482:Relative
6429:(subset)
6421:(subset)
6348:Seminorm
6331:Bilinear
6074:Category
5886:Algebras
5768:Theorems
5725:Complete
5694:Schwartz
5640:glossary
5496:Zonotope
5467:Epigraph
5095:(1979),
4920:(1997).
4898:(1976),
4800:(1980).
4488:18978404
4383:Math. Z.
4354:Zonotope
4315:(a.k.a.
4295:Dilation
4267:See also
3682:: while
3333:For two
3276:OpenSCAD
3216:Metafont
2186:open set
1200:zero set
770:vertices
607:dilation
40:geometry
6845:Sumsets
6754:)
6702:)
6644:K-space
6629:Hilbert
6612:Fréchet
6597:F-space
6572:Brauner
6565:)
6550:)
6532:Asplund
6514:)
6484:)
6404:Bounded
6299:Compact
6284:Bounded
6221: (
5877:Unitary
5857:Nuclear
5842:Compact
5837:Bounded
5832:Adjoint
5806:Min–max
5699:Sobolev
5684:Nuclear
5674:Hilbert
5669:Fréchet
5634: (
5551:Duality
5453:Pseudo-
5427:Ursescu
5324:Pseudo-
5298:Concave
5277:Simplex
5257:Duality
5087:, 2001
4944:1451876
4888:0634800
4824:0439057
4625:1216521
4589:1216521
4563:sumsets
4541:0002009
4533:1968735
4427:(PhD).
4301:Erosion
4244:By the
2125:is the
2039:is the
772:of two
739:Example
611:erosion
46:of two
6766:Webbed
6752:Quasi-
6674:Montel
6664:Mackey
6563:Ultra-
6542:Banach
6450:Radial
6414:Convex
6384:Affine
6326:Linear
6294:Closed
6118:(TVSs)
5852:Normal
5689:Orlicz
5679:Hölder
5659:Banach
5648:Spaces
5636:topics
5534:, and
5505:Series
5422:Simons
5329:Quasi-
5319:Proper
5304:Closed
4959:
4942:
4932:
4906:
4886:
4876:
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4339:Sumset
3666:where
3390:arrows
1840:). If
681:
601:In 2D
170:(also
137:
42:, the
6724:Smith
6709:Riesz
6700:Semi-
6512:Quasi
6506:Polar
5664:Besov
5362:Main
5112:, in
4678:S2CID
4529:JSTOR
4484:S2CID
4454:(PDF)
4399:S2CID
4364:Notes
4121:and
3412:Other
2041:graph
1960:with
178:, or
6343:Norm
6267:form
6255:Maps
6012:(or
5730:Dual
5482:Lens
5436:Sets
5286:Maps
5193:and
5121:and
4930:ISBN
4904:ISBN
4874:ISBN
4810:ISBN
4611:ISBN
4575:ISBN
4464:C-32
4092:For
3987:and
3983:For
3640:>
3424:)).
3404:and
3380:and
3348:and
3340:and
2992:but
2586:are
2541:Conv
2504:Conv
2456:Conv
2431:Conv
2393:Conv
2356:and
1869:and
1290:The
894:and
764:and
609:and
166:The
57:and
48:sets
5536:(Hw
5135:by
5056:doi
5052:240
5031:doi
5027:238
4984:doi
4957:Zbl
4866:doi
4841:doi
4771:doi
4740:doi
4736:238
4709:doi
4705:240
4668:doi
4664:265
4640:at
4521:doi
4476:hdl
4468:doi
4391:doi
4319:or
4125:in
3724:sup
3274:In
3258:In
3214:in
3206:of
3168:180
2593:If
2043:of
1967:If
1962:any
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1544:in
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776:in
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