4951:
612:
604:
685:
can be modified to enclose more area, by "flipping" the concave areas so that they become convex. It can further be shown that any closed curve which is not fully symmetrical can be "tilted" so that it encloses more area. The one shape that is perfectly convex and symmetrical is the circle, although
2651:
590:
and to regions in higher-dimensional spaces. Perhaps the most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water. Namely, a drop will typically assume a symmetric round shape. Since the amount of water in a drop is fixed,
627:
in the plane of fixed perimeter, which curve (if any) maximizes the area of its enclosed region? This question can be shown to be equivalent to the following problem: Among all closed curves in the plane enclosing a fixed area, which curve (if any) minimizes the perimeter?
3492:
Isoperimetric inequalities for graphs relate the size of vertex subsets to the size of their boundary, which is usually measured by the number of edges leaving the subset (edge expansion) or by the number of neighbouring vertices (vertex expansion). For a graph
1637:
307:
3149:
2887:
3786:
3663:
1311:
and that the equality holds if and only if the curve is a circle. There are, in fact, two ways to measure the spherical area enclosed by a simple closed curve, but the inequality is symmetric with the respect to taking the complement.
2305:
5503:. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. BirkhÀuser Boston, Inc., Boston, Massachusetts, 1999.
1074:
4533:
3229:
4676:
2466:
1922:
1869:
4937:
1396:
4193:
4427:
966:
4787:
3392:
4855:
2711:
1682:
353:
2012:
1761:, Sect. 5.2.5) as follows. An extremal set consists of a ball and a "corona" that contributes neither to the volume nor to the surface area. That is, the equality holds for a compact set
639:, in that it can be restated: what is the principle of action which encloses the greatest area, with the greatest economy of effort? The 15th-century philosopher and scientist, Cardinal
1125:
2160:
1521:
1443:
864:
151:
194:
3050:
2064:
1513:
1478:
186:
116:
3670:
2775:
796:(not assumed to be smooth). An elegant direct proof based on comparison of a smooth simple closed curve with an appropriate circle was given by E. Schmidt in 1938. It uses only the
3540:
1306:
3831:
5696:
753:
539:
2751:
2458:
2402:
1735:
668:
Although the circle appears to be an obvious solution to the problem, proving this fact is rather difficult. The first progress toward the solution was made by Swiss geometer
486:
406:
77:
2110:
3305:
then this question generalizes the classical isoperimetric problem to planar regions whose boundary is not necessarily smooth, although the answer turns out to be the same.
5637:(1949). "Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Hugel in der euklidischen und nichteuklidischen Geometrie. II".
4348:
4071:
3283:
2346:
3880:
3437:
2988:
3954:
3927:
2084:
1969:
651:
is generated, to be the most direct reflection, in the realm of sensory impressions, of the process by which the universe is created. German astronomer and astrologer
681:
Steiner begins with some geometric constructions which are easily understood; for example, it can be shown that any closed curve enclosing a region that is not fully
4572:
4240:
4098:
4012:
4302:
435:
2175:
4699:
4322:
4276:
4213:
4032:
3982:
3900:
3851:
3531:
3511:
2910:
2032:
1949:
1819:
1799:
1779:
1706:
1209:
1189:
1165:
1145:
989:
377:
48:
6479:
6557:
583:. However, the first mathematically rigorous proof of this fact was obtained only in the 19th century. Since then, many other proofs have been found.
6574:
3477:
that have strong connectivity properties. Expander constructions have spawned research in pure and applied mathematics, with several applications to
1001:
5096:
1168:
678:. Steiner showed that if a solution existed, then it must be the circle. Steiner's proof was completed later by several other mathematicians.
607:
If a region is not convex, a "dent" in its boundary can be "flipped" to increase the area of the region while keeping the perimeter unchanged.
2929:
4438:
4865:
2646:{\displaystyle \left(\int _{\mathbb {R} ^{n}}|u|^{n/(n-1)}\right)^{(n-1)/n}\leq n^{-1}\omega _{n}^{-1/n}\int _{\mathbb {R} ^{n}}|\nabla u|}
3160:
4580:
1874:
1824:
5882:
5741:
5031:
5204:
6397:
4874:
1332:
6228:
5426:
5366:
4115:
5768:
5335:(in German), 5th edition, completely revised by K. LeichtweiĂ. Die Grundlehren der mathematischen Wissenschaften, Band 1.
4356:
903:
6655:
6389:
4707:
3314:
1749:), the equality holds for a ball only. But in full generality the situation is more complicated. The relevant result of
6175:
6569:
673:
5486:
5464:
5344:
5185:
4985:
4811:
587:
4254:
have the smallest vertex boundary among all sets of a given size. Hamming balls are sets that contain all points of
2659:
6526:
6516:
1645:
316:
5700:
1974:
1632:{\displaystyle \operatorname {per} (S)\geq n\operatorname {vol} (S)^{(n-1)/n}\,\operatorname {vol} (B_{1})^{1/n},}
6650:
6645:
6640:
6326:
6235:
5999:
3478:
302:{\displaystyle \operatorname {per} (S)\geq n\operatorname {vol} (S)^{(n-1)/n}\,\operatorname {vol} (B_{1})^{1/n}}
3144:{\displaystyle \mu ^{+}(A)=\liminf _{\varepsilon \to 0+}{\frac {\mu (A_{\varepsilon })-\mu (A)}{\varepsilon }},}
1082:
5496:
2913:
2882:{\displaystyle \operatorname {per} (S)\geq n\operatorname {vol} (S)^{(n-1)/n}\operatorname {vol} (B_{1})^{1/n}}
2130:
1413:
821:
121:
5855:
5377:
6564:
6511:
6405:
6311:
5387:
5000:
4964:
2724:
are complete simply connected manifolds with nonpositive curvature. Thus they generalize the
Euclidean space
2037:
1486:
1451:
805:
159:
89:
5595:
ZwierzyĆski, MichaĆ (2016). "The improved isoperimetric inequality and the Wigner caustic of planar ovals".
5127:
ZwierzyĆski, MichaĆ (2016). "The improved isoperimetric inequality and the Wigner caustic of planar ovals".
6430:
6410:
6374:
6298:
6018:
5734:
1257:
3794:
1316:
6552:
6331:
6293:
6245:
5382:
1928:
686:
this, in itself, does not represent a rigorous proof of the isoperimetric theorem (see external links).
6635:
6457:
6425:
6415:
6336:
6303:
5934:
5843:
5443:. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 3. Berlin: 1. Verlag von Julius Springer.
2948:. However, the isoperimetric problem can be formulated in much greater generality, using the notion of
2758:
815:
is defined as the ratio of its area and that of the circle having the same perimeter. This is equal to
716:
502:
2727:
2434:
2378:
1711:
462:
382:
53:
6474:
6379:
6155:
6083:
2089:
632:
6464:
5307:
Dragutin Svrtan and Darko Veljan, "Non-Euclidean
Versions of Some Classical Triangle Inequalities",
615:
An elongated shape can be made more round while keeping its perimeter fixed and increasing its area.
5993:
5924:
5005:
695:
657:
568:
27:
5860:
5117:, (1838), pp. 281–296; and Gesammelte Werke Vol. 2, pp. 77–91, Reimer, Berlin, (1882).
4868:, by a stronger inequality which has also been called the isoperimetric inequality for triangles:
4327:
595:
forces the drop into a shape which minimizes the surface area of the drop, namely a round sphere.
6316:
6074:
6034:
5727:
5070:
4990:
4037:
3252:
2313:
1212:
3856:
3404:
2955:
6599:
6499:
6321:
6043:
5889:
5205:
http://www.ams.org/journals/bull/1978-84-06/S0002-9904-1978-14553-4/S0002-9904-1978-14553-4.pdf
3932:
3905:
3781:{\displaystyle \Phi _{V}(G,k)=\min _{S\subseteq V}\left\{|\Gamma (S)\setminus S|:|S|=k\right\}}
2069:
1954:
5710:
3658:{\displaystyle \Phi _{E}(G,k)=\min _{S\subseteq V}\left\{|E(S,{\overline {S}})|:|S|=k\right\}}
451:
6160:
6113:
6108:
6103:
5945:
5828:
5786:
5203:. "The Isoperimetric Inequality." Bulletin of the American Mathematical Society. 84.6 (1978)
4975:
3486:
2427:-dimensional isoperimetric inequality is equivalent (for sufficiently smooth domains) to the
5529:
4104:
is exactly one. The following are the isoperimetric inequalities for the
Boolean hypercube.
2940:
Most of the work on isoperimetric problem has been done in the context of smooth regions in
6469:
6435:
6343:
6053:
6008:
5850:
5773:
5357:
Combinatorics: set systems, hypergraphs, families of vectors, and combinatorial probability
4861:
4541:
4218:
4076:
3990:
3003:
2300:{\displaystyle n\,\omega _{n}^{1/n}L^{n}({\bar {S}})^{(n-1)/n}\leq M_{*}^{n-1}(\partial S)}
16:
Geometric inequality which sets a lower bound on the surface area of a set given its volume
5418:
8:
6452:
6442:
6288:
6252:
6078:
5807:
5764:
4995:
4281:
3298:
2945:
874:
655:
invoked the isoperimetric principle in discussing the morphology of the solar system, in
556:
414:
6130:
5298:(R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
586:
The isoperimetric problem has been extended in multiple ways, for example, to curves on
6604:
6364:
6349:
6048:
5929:
5907:
5705:
5676:
5622:
5604:
5453:
5350:
5312:
5154:
5136:
5051:
4956:
4684:
4307:
4261:
4198:
4017:
3967:
3885:
3836:
3516:
3496:
2895:
2428:
2017:
1934:
1804:
1784:
1764:
1742:
1691:
1194:
1174:
1150:
1130:
974:
801:
362:
33:
6521:
6257:
6218:
6213:
6120:
6038:
5823:
5796:
5680:
5626:
5482:
5460:
5422:
5395:
5362:
5340:
5181:
5158:
5090:
4969:
4950:
3026:
2721:
2417:
2353:
793:
5585:
5568:
5415:
An
Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem
611:
603:
6538:
6447:
6223:
6208:
6198:
6183:
6150:
6145:
6135:
6013:
5988:
5803:
5668:
5646:
5614:
5580:
5544:
5328:
5146:
5043:
4101:
3482:
3302:
2921:
2766:
2167:
2163:
640:
563:
asks for a region of the maximal area bounded by a straight line and a curvilinear
5549:
5355:
6614:
6594:
6369:
6267:
6262:
6240:
6098:
6063:
5983:
5877:
5474:
5336:
5200:
3470:
2941:
2409:
694:
The solution to the isoperimetric problem is usually expressed in the form of an
652:
592:
5412:
623:
dates back to antiquity. The problem can be stated as follows: Among all closed
6504:
6359:
6354:
6165:
6140:
6093:
6023:
6003:
5963:
5953:
5750:
5634:
5618:
5448:
5436:
5150:
4980:
4255:
3460:
3444:
1746:
1069:{\displaystyle L^{2}\geqslant 4\pi A+8\pi \left|{\widetilde {A}}_{0.5}\right|,}
789:
759:
580:
564:
2753:, which is a Hadamard manifold with curvature zero. In 1970's and early 80's,
6629:
6609:
6272:
6193:
6188:
6088:
6058:
6028:
5978:
5973:
5968:
5958:
5872:
5791:
5650:
5525:
5507:
3014:
2925:
2754:
785:
669:
438:
3447:
and for special classes of
Riemannian manifolds (where usually only regions
2917:
6203:
6125:
5865:
5714:
5010:
3474:
3466:
3440:
2999:
1446:
784:
Dozens of proofs of the isoperimetric inequality have been found. In 1902,
552:
442:
80:
5902:
4701:. Then the above implies that the exact vertex isoperimetric parameter is
6068:
5521:
3902:. The isoperimetric problem consists of understanding how the parameters
2762:
1685:
356:
5459:. Moscow, Idaho: L. Boron, C. Christenson and B. Smith. BCS Associates.
5055:
5912:
5672:
3030:
1738:
1319:(1919) who also extended it to higher dimensions and general surfaces.
797:
682:
4972:: isoperimetric problem is a zero wind speed case of Chaplygin problem
4528:{\displaystyle |S\cup \Gamma (S)|\geq \sum _{i=0}^{r+1}{d \choose i}.}
3533:, the following are two standard isoperimetric parameters for graphs.
758:
and that the equality holds if and only if the curve is a circle. The
5894:
5838:
5833:
5413:
Capogna, Luca; Donatella
Danielli; Scott Pauls; Jeremy Tyson (2007).
5173:
5047:
3985:
3959:
2372:
456:
84:
1410:
has the smallest surface area per given volume. Given a bounded set
5919:
5778:
5609:
5141:
3224:{\displaystyle A_{\varepsilon }=\{x\in X|d(x,A)\leq \varepsilon \}}
644:
572:
24:
4671:{\displaystyle k={d \choose 0}+{d \choose 1}+\dots +{d \choose r}}
5719:
4797:
The isoperimetric inequality for triangles in terms of perimeter
3042:
2127:, §3.2.43), the isoperimetric inequality states that for any set
636:
5180:. Modern BirkhÀuser Classics. Dordrecht: Springer. p. 519.
2924:
at the time. In dimensions 3 and 4 the conjecture was proved by
1917:{\displaystyle \operatorname {per} (B)=\operatorname {per} (S).}
1864:{\displaystyle \operatorname {vol} (B)=\operatorname {vol} (S)}
1481:
1407:
1228:
648:
576:
154:
4014:
is the graph whose vertices are all
Boolean vectors of length
3469:, isoperimetric inequalities are at the heart of the study of
2916:. In dimension 2 this had already been established in 1926 by
544:
and that equality holds if and only if the curve is a circle.
624:
5558:
Leader, Imre (1991). "Discrete isoperimetric inequalities".
446:
5501:
Metric structures for
Riemannian and non-Riemannian spaces
5294:
Chakerian, G. D. "A Distorted View of
Geometry." Ch. 7 in
5178:
Metric
Structures for Riemannian and Non-Riemannian Spaces
4100:
if they are equal up to a single bit flip, that is, their
575:. The solution to the isoperimetric problem is given by a
4792:
598:
437:, the isoperimetric inequality relates the square of the
5706:
Treiberg: Several proofs of the isoperimetric inequality
5499:: "Paul Levy's isoperimetric inequality". Appendix C in
4932:{\displaystyle T\leq {\frac {\sqrt {3}}{4}}(abc)^{2/3}.}
2769:
conjectured that the Euclidean isoperimetric inequality
1391:{\displaystyle L^{2}\geq 4\pi A-{\frac {A^{2}}{R^{2}}}.}
800:
formula, expression for the area of a plane region from
567:
whose endpoints belong to that line. It is named after
5447:
5435:
4188:{\displaystyle \Phi _{E}(Q_{d},k)\geq k(d-\log _{2}k)}
4112:
The edge isoperimetric inequality of the hypercube is
1191:, respectively, and the equality holds if and only if
488:, the isoperimetric inequality states, for the length
5659:
Baebler, F. (1957). "Zum isoperimetrischen Problem".
5512:
Vorlesungen ĂŒber Inhalt, OberflĂ€che und Isoperimetrie
4877:
4814:
4710:
4687:
4583:
4544:
4441:
4359:
4330:
4310:
4284:
4264:
4221:
4201:
4118:
4079:
4040:
4020:
3993:
3970:
3935:
3908:
3888:
3882:
denotes the set of vertices that have a neighbour in
3859:
3839:
3797:
3673:
3543:
3519:
3499:
3407:
3317:
3255:
3163:
3053:
2958:
2912:
in Hadamard manifolds, which has become known as the
2898:
2778:
2730:
2662:
2469:
2437:
2381:
2316:
2178:
2133:
2092:
2072:
2040:
2020:
1977:
1957:
1937:
1877:
1827:
1807:
1787:
1767:
1714:
1694:
1648:
1524:
1489:
1454:
1416:
1335:
1260:
1197:
1177:
1153:
1133:
1085:
1004:
977:
906:
824:
719:
505:
465:
417:
385:
365:
319:
197:
162:
124:
92:
56:
36:
5519:
5313:
http://forumgeom.fau.edu/FG2012volume12/FG201217.pdf
5283:
5243:
5227:
5215:
4946:
4422:{\displaystyle |S|\geq \sum _{i=0}^{r}{d \choose i}}
2014:. By taking BrunnâMinkowski inequality to the power
1737:. Under additional restrictions on the set (such as
961:{\displaystyle Q_{n}={\frac {\pi }{n\tan(\pi /n)}}.}
30:
involving the perimeter of a set and its volume. In
4782:{\displaystyle \Phi _{V}(Q_{d},k)={d \choose r+1}.}
4195:. This bound is tight, as is witnessed by each set
5452:
5354:
5174:"Appendix C. Paul Levy's Isoperimetric Inequality"
4931:
4849:
4781:
4693:
4670:
4566:
4527:
4421:
4342:
4316:
4296:
4270:
4245:
4234:
4207:
4187:
4092:
4065:
4026:
4006:
3976:
3960:Example: Isoperimetric inequalities for hypercubes
3948:
3921:
3894:
3874:
3845:
3825:
3780:
3657:
3525:
3505:
3431:
3387:{\displaystyle I(a)=\inf\{\mu ^{+}(A)|\mu (A)=a\}}
3386:
3277:
3223:
3143:
2982:
2904:
2881:
2745:
2705:
2645:
2452:
2396:
2340:
2299:
2154:
2104:
2078:
2058:
2026:
2006:
1963:
1943:
1927:The proof of the inequality follows directly from
1916:
1863:
1813:
1793:
1773:
1729:
1700:
1676:
1631:
1507:
1472:
1437:
1390:
1300:
1203:
1183:
1159:
1139:
1119:
1068:
991:be a smooth regular convex closed curve. Then the
983:
960:
858:
747:
533:
480:
429:
400:
371:
347:
301:
180:
145:
110:
71:
42:
5111:Einfacher Beweis der isoperimetrischen HauptsÀtze
4770:
4749:
4662:
4649:
4631:
4618:
4606:
4593:
4516:
4503:
4413:
4400:
6627:
4107:
3703:
3573:
3333:
3077:
774:, so both sides of the inequality are equal to 4
4850:{\displaystyle p^{2}\geq 12{\sqrt {3}}\cdot T,}
4073:. Two such vectors are connected by an edge in
3439:. Isoperimetric profiles have been studied for
5560:Proceedings of Symposia in Applied Mathematics
5069:Olmo, Carlos BeltrĂĄn, Irene (4 January 2021).
5029:
4215:that is the set of vertices of any subcube of
2706:{\displaystyle u\in W^{1,1}(\mathbb {R} ^{n})}
672:in 1838, using a geometric method later named
5735:
5537:Bulletin of the American Mathematical Society
1677:{\displaystyle B_{1}\subset \mathbb {R} ^{n}}
1322:In the more general case of arbitrary radius
348:{\displaystyle B_{1}\subset \mathbb {R} ^{n}}
6480:RieszâMarkovâKakutani representation theorem
5095:: CS1 maint: multiple names: authors list (
5032:"The Evolution of the Isoperimetric Problem"
4278:and no points of Hamming weight larger than
4054:
4041:
3381:
3336:
3218:
3177:
2935:
2007:{\displaystyle B_{\epsilon }=\epsilon B_{1}}
631:This problem is conceptually related to the
559:has a specified length. The closely related
496:of the planar region that it encloses, that
5594:
5171:
5126:
1406:The isoperimetric inequality states that a
869:and the isoperimetric inequality says that
706:of the planar region that it encloses. The
571:, the legendary founder and first queen of
6575:Vitale's random BrunnâMinkowski inequality
5742:
5728:
1924:For example, the "corona" may be a curve.
1120:{\displaystyle L,A,{\widetilde {A}}_{0.5}}
5608:
5584:
5548:
5288:
5140:
2733:
2690:
2615:
2483:
2440:
2384:
2182:
2155:{\displaystyle S\subset \mathbb {R} ^{n}}
2142:
1717:
1664:
1588:
1438:{\displaystyle S\subset \mathbb {R} ^{n}}
1425:
859:{\displaystyle Q={\frac {4\pi A}{L^{2}}}}
468:
388:
335:
261:
146:{\displaystyle S\subset \mathbb {R} ^{n}}
133:
59:
5566:
5530:"Expander graphs and their applications"
5506:
5349:
5259:
5239:
2716:
2113:
1758:
893:The isoperimetric quotient of a regular
770:and the circumference of the circle is 2
610:
602:
5658:
5633:
5473:
5393:
5255:
5172:Gromov, Mikhail; Pansu, Pierre (2006).
3956:behave for natural families of graphs.
3451:with regular boundary are considered).
2124:
2117:
2059:{\displaystyle \operatorname {vol} (S)}
1754:
1753:, Sect. 20.7) (for a simpler proof see
1750:
1508:{\displaystyle \operatorname {vol} (S)}
1473:{\displaystyle \operatorname {per} (S)}
181:{\displaystyle \operatorname {vol} (S)}
111:{\displaystyle \operatorname {per} (S)}
6628:
5557:
5375:
5271:
4793:Isoperimetric inequality for triangles
4538:As a special case, consider set sizes
1515:, the isoperimetric inequality states
599:The isoperimetric problem in the plane
5723:
2412:, then the Minkowski content is the (
1401:
1301:{\displaystyle L^{2}\geq A(4\pi -A),}
6588:Applications & related
5697:History of the Isoperimetric Problem
5284:Hoory, Linial & Widgerson (2006)
5244:Hoory, Linial & Widgerson (2006)
5228:Hoory, Linial & Widgerson (2006)
5216:Hoory, Linial & Widgerson (2006)
5068:
4324:. This theorem implies that any set
3826:{\displaystyle E(S,{\overline {S}})}
3667:The vertex isoperimetric parameter:
1147:, the area of the region bounded by
5339:, New York Heidelberg Berlin, 1973
2364:-dimensional Lebesgue measure, and
555:of the largest possible area whose
13:
5749:
4753:
4712:
4653:
4622:
4597:
4507:
4453:
4404:
4120:
3937:
3910:
3860:
3728:
3675:
3545:
3537:The edge isoperimetric parameter:
2632:
2288:
2066:from both sides, dividing them by
1315:This inequality was discovered by
1249:spherical isoperimetric inequality
788:published a short proof using the
14:
6667:
5690:
4986:Gaussian isoperimetric inequality
3833:denotes the set of edges leaving
3740:
993:improved isoperimetric inequality
748:{\displaystyle 4\pi A\leq L^{2},}
534:{\displaystyle L^{2}\geq 4\pi A,}
455:literally means "having the same
6517:Lebesgue differentiation theorem
6398:Carathéodory's extension theorem
5333:Elementare Differentialgeometrie
4949:
3301:with the usual distance and the
2746:{\displaystyle \mathbb {R} ^{n}}
2453:{\displaystyle \mathbb {R} ^{n}}
2397:{\displaystyle \mathbb {R} ^{n}}
1730:{\displaystyle \mathbb {R} ^{n}}
663:The Sacred Mystery of the Cosmos
481:{\displaystyle \mathbb {R} ^{2}}
401:{\displaystyle \mathbb {R} ^{n}}
79:the inequality lower bounds the
72:{\displaystyle \mathbb {R} ^{n}}
5586:10.1090/S0002-9904-1978-14553-4
5562:. Vol. 44. pp. 57â80.
5301:
5276:
5264:
5248:
4246:Vertex isoperimetric inequality
2105:{\displaystyle \epsilon \to 0.}
702:of a closed curve and the area
647:action, the process by which a
492:of a closed curve and the area
449:of a plane region it encloses.
359:. The equality holds only when
5569:"The isoperimetric inequality"
5361:. Cambridge University Press.
5232:
5220:
5209:
5194:
5165:
5120:
5103:
5062:
5023:
4909:
4896:
4740:
4721:
4560:
4552:
4466:
4462:
4456:
4443:
4369:
4361:
4182:
4157:
4148:
4129:
3869:
3863:
3820:
3801:
3763:
3755:
3747:
3737:
3731:
3724:
3696:
3684:
3640:
3632:
3624:
3620:
3601:
3594:
3566:
3554:
3426:
3408:
3372:
3366:
3359:
3355:
3349:
3327:
3321:
3272:
3266:
3209:
3197:
3190:
3129:
3123:
3114:
3101:
3084:
3070:
3064:
2977:
2959:
2862:
2848:
2829:
2817:
2813:
2806:
2791:
2785:
2700:
2685:
2639:
2628:
2553:
2541:
2530:
2518:
2505:
2496:
2294:
2285:
2248:
2236:
2232:
2225:
2216:
2096:
2053:
2047:
1908:
1902:
1890:
1884:
1858:
1852:
1840:
1834:
1609:
1595:
1575:
1563:
1559:
1552:
1537:
1531:
1502:
1496:
1467:
1461:
1292:
1277:
1227:be a simple closed curve on a
1218:
949:
935:
811:For a given closed curve, the
282:
268:
248:
236:
232:
225:
210:
204:
175:
169:
105:
99:
1:
5550:10.1090/S0273-0979-06-01126-8
5321:
5001:List of triangle inequalities
4108:Edge isoperimetric inequality
3454:
3245:The isoperimetric problem in
1167:and the oriented area of the
689:
4343:{\displaystyle S\subseteq V}
3815:
3615:
3401:of the metric measure space
7:
6570:PrĂ©kopaâLeindler inequality
5441:Theorie der konvexen Körper
5383:Encyclopedia of Mathematics
5226:Definitions 4.2 and 4.3 of
4942:
4864:. This is implied, via the
4250:Harper's theorem says that
4066:{\displaystyle \{0,1\}^{d}}
3278:{\displaystyle \mu ^{+}(A)}
2341:{\displaystyle M_{*}^{n-1}}
10:
6672:
6656:Theorems in measure theory
6512:Lebesgue's density theorem
5619:10.1016/j.jmaa.2016.05.016
5451:; Bonnesen, Tommy (1987).
5439:; Bonnesen, Tommy (1934).
5378:"Isoperimetric inequality"
5151:10.1016/j.jmaa.2016.05.016
3875:{\displaystyle \Gamma (S)}
3458:
3432:{\displaystyle (X,\mu ,d)}
2983:{\displaystyle (X,\mu ,d)}
2914:CartanâHadamard conjecture
2086:, and taking the limit as
1929:BrunnâMinkowski inequality
1688:. The equality holds when
792:that applies to arbitrary
6587:
6565:MinkowskiâSteiner formula
6535:
6495:
6488:
6388:
6380:Projection-valued measure
6281:
6174:
5943:
5816:
5757:
5567:Osserman, Robert (1978).
4965:BlaschkeâLebesgue theorem
3949:{\displaystyle \Phi _{V}}
3922:{\displaystyle \Phi _{E}}
2936:In a metric measure space
2079:{\displaystyle \epsilon }
1964:{\displaystyle \epsilon }
806:CauchyâSchwarz inequality
633:principle of least action
579:and was known already in
6548:Isoperimetric inequality
6527:VitaliâHahnâSaks theorem
5856:Carathéodory's criterion
5651:10.1002/mana.19490020308
5479:Geometric measure theory
5016:
5006:Planar separator theorem
2944:, or more generally, in
708:isoperimetric inequality
698:that relates the length
658:Mysterium Cosmographicum
21:isoperimetric inequality
6553:BrunnâMinkowski theorem
6422:Decomposition theorems
5455:Theory of convex bodies
5394:Calabro, Chris (2004).
5113:, J. reine angew Math.
5030:BlÄsjö, Viktor (2005).
4991:Isoperimetric dimension
2920:, who was a student of
2892:holds for bounded sets
2460:with optimal constant:
1951:and a ball with radius
1801:contains a closed ball
1231:of radius 1. Denote by
1213:curve of constant width
873:†1. Equivalently, the
6651:Multivariable calculus
6646:Geometric inequalities
6641:Calculus of variations
6600:Descriptive set theory
6500:Disintegration theorem
5935:Universally measurable
4933:
4860:with equality for the
4851:
4783:
4695:
4672:
4568:
4529:
4499:
4423:
4396:
4344:
4318:
4298:
4272:
4236:
4209:
4189:
4094:
4067:
4028:
4008:
3978:
3950:
3923:
3896:
3876:
3847:
3827:
3782:
3659:
3527:
3507:
3487:error-correcting codes
3433:
3388:
3279:
3225:
3145:
2984:
2932:in 1984 respectively.
2906:
2883:
2747:
2707:
2647:
2454:
2398:
2342:
2301:
2156:
2106:
2080:
2060:
2028:
2008:
1965:
1945:
1918:
1865:
1815:
1795:
1775:
1731:
1702:
1678:
1633:
1509:
1474:
1439:
1392:
1302:
1205:
1185:
1161:
1141:
1121:
1070:
985:
962:
860:
813:isoperimetric quotient
749:
675:Steiner symmetrisation
616:
608:
535:
482:
431:
411:On a plane, i.e. when
402:
373:
349:
303:
182:
147:
112:
73:
44:
6402:Convergence theorems
5861:Cylindrical Ï-algebra
5711:Isoperimetric Theorem
5573:Bull. Amer. Math. Soc
5071:"Sobre mates y mitos"
4976:Curve-shortening flow
4934:
4852:
4784:
4696:
4673:
4569:
4567:{\displaystyle k=|S|}
4530:
4473:
4424:
4376:
4345:
4319:
4299:
4273:
4237:
4235:{\displaystyle Q_{d}}
4210:
4190:
4095:
4093:{\displaystyle Q_{d}}
4068:
4029:
4009:
4007:{\displaystyle Q_{d}}
3979:
3951:
3924:
3897:
3877:
3848:
3828:
3783:
3660:
3528:
3508:
3434:
3399:isoperimetric profile
3389:
3280:
3226:
3146:
2985:
2907:
2884:
2748:
2717:In Hadamard manifolds
2708:
2648:
2455:
2404:. If the boundary of
2399:
2371:is the volume of the
2343:
2302:
2157:
2107:
2081:
2061:
2029:
2009:
1966:
1946:
1919:
1866:
1816:
1796:
1776:
1732:
1703:
1679:
1634:
1510:
1475:
1440:
1393:
1303:
1243:the area enclosed by
1206:
1186:
1162:
1142:
1127:denote the length of
1122:
1071:
995:states the following
986:
963:
861:
750:
621:isoperimetric problem
614:
606:
549:isoperimetric problem
536:
483:
432:
403:
374:
350:
304:
183:
148:
113:
74:
45:
6470:Minkowski inequality
6344:Cylinder set measure
6229:Infinite-dimensional
5844:equivalence relation
5774:Lebesgue integration
4875:
4862:equilateral triangle
4812:
4708:
4685:
4581:
4542:
4439:
4357:
4328:
4308:
4282:
4262:
4219:
4199:
4116:
4077:
4038:
4018:
3991:
3968:
3933:
3906:
3886:
3857:
3837:
3795:
3671:
3541:
3517:
3497:
3485:, and the theory of
3405:
3315:
3253:
3161:
3051:
2992:metric measure space
2956:
2946:Riemannian manifolds
2896:
2776:
2728:
2660:
2467:
2435:
2379:
2314:
2176:
2131:
2123:In full generality (
2090:
2070:
2038:
2018:
1975:
1955:
1935:
1875:
1825:
1805:
1785:
1765:
1712:
1692:
1646:
1522:
1487:
1452:
1414:
1333:
1258:
1195:
1175:
1151:
1131:
1083:
1002:
975:
904:
822:
717:
503:
463:
415:
383:
363:
317:
195:
160:
122:
90:
54:
34:
19:In mathematics, the
6465:Hölder's inequality
6327:of random variables
6289:Measurable function
6176:Particular measures
5765:Absolute continuity
5661:Arch. Math. (Basel)
5597:J. Math. Anal. Appl
5481:. Springer-Verlag.
5311:12, 2012, 197â209.
5309:Forum Geometricorum
5129:J. Math. Anal. Appl
5036:Amer. Math. Monthly
4996:Isoperimetric point
4297:{\displaystyle r+1}
4034:, that is, the set
3481:, design of robust
3249:asks how small can
2607:
2337:
2284:
2205:
1326:, it is known that
875:isoperimetric ratio
459:". Specifically in
430:{\displaystyle n=2}
50:-dimensional space
6605:Probability theory
5930:Transverse measure
5908:Non-measurable set
5890:Locally measurable
5673:10.1007/BF01898439
5514:. Springer-Verlag.
5396:"Harper's Theorem"
5296:Mathematical Plums
4957:Mathematics portal
4929:
4847:
4779:
4691:
4668:
4564:
4525:
4419:
4340:
4314:
4294:
4268:
4232:
4205:
4185:
4090:
4063:
4024:
4004:
3974:
3946:
3919:
3892:
3872:
3843:
3823:
3778:
3717:
3655:
3587:
3523:
3503:
3429:
3384:
3275:
3221:
3141:
3094:
3041:is defined as the
2980:
2902:
2879:
2743:
2722:Hadamard manifolds
2703:
2643:
2582:
2450:
2429:Sobolev inequality
2394:
2338:
2317:
2297:
2264:
2183:
2152:
2102:
2076:
2056:
2024:
2004:
1961:
1941:
1914:
1861:
1811:
1791:
1771:
1757:) is clarified in
1727:
1698:
1674:
1629:
1505:
1470:
1435:
1402:In Euclidean space
1388:
1298:
1201:
1181:
1157:
1137:
1117:
1066:
981:
958:
856:
794:rectifiable curves
745:
617:
609:
551:is to determine a
531:
478:
427:
398:
369:
345:
299:
178:
143:
108:
69:
40:
6636:Analytic geometry
6623:
6622:
6583:
6582:
6312:almost everywhere
6258:Spherical measure
6156:Strictly positive
6084:Projection-valued
5824:Almost everywhere
5797:Probability space
5428:978-3-7643-8132-5
5419:BirkhÀuser Verlag
5368:978-0-521-33703-8
5242:and Section 4 in
4970:Chaplygin problem
4894:
4890:
4836:
4768:
4694:{\displaystyle r}
4681:for some integer
4660:
4629:
4604:
4514:
4411:
4317:{\displaystyle r}
4304:for some integer
4271:{\displaystyle r}
4208:{\displaystyle S}
4027:{\displaystyle d}
3977:{\displaystyle d}
3895:{\displaystyle S}
3846:{\displaystyle S}
3818:
3702:
3618:
3572:
3526:{\displaystyle k}
3506:{\displaystyle G}
3483:computer networks
3479:complexity theory
3136:
3076:
3027:Minkowski content
2950:Minkowski content
2905:{\displaystyle S}
2418:Hausdorff measure
2354:Minkowski content
2228:
2027:{\displaystyle n}
1944:{\displaystyle S}
1814:{\displaystyle B}
1794:{\displaystyle S}
1774:{\displaystyle S}
1701:{\displaystyle S}
1383:
1204:{\displaystyle C}
1184:{\displaystyle C}
1160:{\displaystyle C}
1140:{\displaystyle C}
1108:
1050:
984:{\displaystyle C}
953:
890:for every curve.
854:
372:{\displaystyle S}
43:{\displaystyle n}
6663:
6558:Milman's reverse
6541:
6539:Lebesgue measure
6493:
6492:
5897:
5883:infimum/supremum
5804:Measurable space
5744:
5737:
5730:
5721:
5720:
5684:
5654:
5645:(3â4): 171â244.
5630:
5612:
5590:
5588:
5579:(6): 1182â1238.
5563:
5554:
5552:
5534:
5515:
5492:
5475:Federer, Herbert
5470:
5458:
5444:
5432:
5409:
5407:
5405:
5400:
5390:
5376:Burago (2001) ,
5372:
5360:
5331:and LeichtweiĂ,
5315:
5305:
5299:
5292:
5286:
5280:
5274:
5268:
5262:
5252:
5246:
5236:
5230:
5224:
5218:
5213:
5207:
5201:Osserman, Robert
5198:
5192:
5191:
5169:
5163:
5162:
5144:
5124:
5118:
5107:
5101:
5100:
5094:
5086:
5084:
5082:
5066:
5060:
5059:
5048:10.2307/30037526
5027:
4959:
4954:
4953:
4938:
4936:
4935:
4930:
4925:
4924:
4920:
4895:
4886:
4885:
4866:AMâGM inequality
4856:
4854:
4853:
4848:
4837:
4832:
4824:
4823:
4788:
4786:
4785:
4780:
4775:
4774:
4773:
4767:
4752:
4733:
4732:
4720:
4719:
4700:
4698:
4697:
4692:
4677:
4675:
4674:
4669:
4667:
4666:
4665:
4652:
4636:
4635:
4634:
4621:
4611:
4610:
4609:
4596:
4573:
4571:
4570:
4565:
4563:
4555:
4534:
4532:
4531:
4526:
4521:
4520:
4519:
4506:
4498:
4487:
4469:
4446:
4428:
4426:
4425:
4420:
4418:
4417:
4416:
4403:
4395:
4390:
4372:
4364:
4349:
4347:
4346:
4341:
4323:
4321:
4320:
4315:
4303:
4301:
4300:
4295:
4277:
4275:
4274:
4269:
4241:
4239:
4238:
4233:
4231:
4230:
4214:
4212:
4211:
4206:
4194:
4192:
4191:
4186:
4175:
4174:
4141:
4140:
4128:
4127:
4102:Hamming distance
4099:
4097:
4096:
4091:
4089:
4088:
4072:
4070:
4069:
4064:
4062:
4061:
4033:
4031:
4030:
4025:
4013:
4011:
4010:
4005:
4003:
4002:
3983:
3981:
3980:
3975:
3955:
3953:
3952:
3947:
3945:
3944:
3928:
3926:
3925:
3920:
3918:
3917:
3901:
3899:
3898:
3893:
3881:
3879:
3878:
3873:
3852:
3850:
3849:
3844:
3832:
3830:
3829:
3824:
3819:
3811:
3787:
3785:
3784:
3779:
3777:
3773:
3766:
3758:
3750:
3727:
3716:
3683:
3682:
3664:
3662:
3661:
3656:
3654:
3650:
3643:
3635:
3627:
3619:
3611:
3597:
3586:
3553:
3552:
3532:
3530:
3529:
3524:
3512:
3510:
3509:
3504:
3438:
3436:
3435:
3430:
3393:
3391:
3390:
3385:
3362:
3348:
3347:
3303:Lebesgue measure
3284:
3282:
3281:
3276:
3265:
3264:
3230:
3228:
3227:
3222:
3193:
3173:
3172:
3150:
3148:
3147:
3142:
3137:
3132:
3113:
3112:
3096:
3093:
3063:
3062:
3023:boundary measure
2989:
2987:
2986:
2981:
2942:Euclidean spaces
2911:
2909:
2908:
2903:
2888:
2886:
2885:
2880:
2878:
2877:
2873:
2860:
2859:
2841:
2840:
2836:
2767:Viktor Zalgaller
2752:
2750:
2749:
2744:
2742:
2741:
2736:
2712:
2710:
2709:
2704:
2699:
2698:
2693:
2684:
2683:
2652:
2650:
2649:
2644:
2642:
2631:
2626:
2625:
2624:
2623:
2618:
2606:
2602:
2590:
2581:
2580:
2565:
2564:
2560:
2539:
2535:
2534:
2533:
2517:
2508:
2499:
2494:
2493:
2492:
2491:
2486:
2459:
2457:
2456:
2451:
2449:
2448:
2443:
2416:-1)-dimensional
2403:
2401:
2400:
2395:
2393:
2392:
2387:
2352:-1)-dimensional
2347:
2345:
2344:
2339:
2336:
2325:
2306:
2304:
2303:
2298:
2283:
2272:
2260:
2259:
2255:
2230:
2229:
2221:
2215:
2214:
2204:
2200:
2191:
2168:Lebesgue measure
2161:
2159:
2158:
2153:
2151:
2150:
2145:
2111:
2109:
2108:
2103:
2085:
2083:
2082:
2077:
2065:
2063:
2062:
2057:
2033:
2031:
2030:
2025:
2013:
2011:
2010:
2005:
2003:
2002:
1987:
1986:
1970:
1968:
1967:
1962:
1950:
1948:
1947:
1942:
1923:
1921:
1920:
1915:
1870:
1868:
1867:
1862:
1820:
1818:
1817:
1812:
1800:
1798:
1797:
1792:
1780:
1778:
1777:
1772:
1736:
1734:
1733:
1728:
1726:
1725:
1720:
1707:
1705:
1704:
1699:
1683:
1681:
1680:
1675:
1673:
1672:
1667:
1658:
1657:
1638:
1636:
1635:
1630:
1625:
1624:
1620:
1607:
1606:
1587:
1586:
1582:
1514:
1512:
1511:
1506:
1479:
1477:
1476:
1471:
1444:
1442:
1441:
1436:
1434:
1433:
1428:
1397:
1395:
1394:
1389:
1384:
1382:
1381:
1372:
1371:
1362:
1345:
1344:
1307:
1305:
1304:
1299:
1270:
1269:
1210:
1208:
1207:
1202:
1190:
1188:
1187:
1182:
1166:
1164:
1163:
1158:
1146:
1144:
1143:
1138:
1126:
1124:
1123:
1118:
1116:
1115:
1110:
1109:
1101:
1075:
1073:
1072:
1067:
1062:
1058:
1057:
1052:
1051:
1043:
1014:
1013:
990:
988:
987:
982:
967:
965:
964:
959:
954:
952:
945:
921:
916:
915:
889:
885:
865:
863:
862:
857:
855:
853:
852:
843:
832:
754:
752:
751:
746:
741:
740:
641:Nicholas of Cusa
540:
538:
537:
532:
515:
514:
487:
485:
484:
479:
477:
476:
471:
436:
434:
433:
428:
407:
405:
404:
399:
397:
396:
391:
378:
376:
375:
370:
354:
352:
351:
346:
344:
343:
338:
329:
328:
308:
306:
305:
300:
298:
297:
293:
280:
279:
260:
259:
255:
187:
185:
184:
179:
152:
150:
149:
144:
142:
141:
136:
117:
115:
114:
109:
78:
76:
75:
70:
68:
67:
62:
49:
47:
46:
41:
6671:
6670:
6666:
6665:
6664:
6662:
6661:
6660:
6626:
6625:
6624:
6619:
6615:Spectral theory
6595:Convex analysis
6579:
6536:
6531:
6484:
6384:
6332:in distribution
6277:
6170:
6000:Logarithmically
5939:
5895:
5878:Essential range
5812:
5753:
5748:
5693:
5688:
5635:Schmidt, Erhard
5532:
5520:Hoory, Shlomo;
5489:
5467:
5449:Fenchel, Werner
5437:Fenchel, Werner
5429:
5403:
5401:
5398:
5369:
5337:Springer-Verlag
5324:
5319:
5318:
5306:
5302:
5293:
5289:
5282:Also stated in
5281:
5277:
5269:
5265:
5260:BollobĂĄs (1986)
5253:
5249:
5240:BollobĂĄs (1986)
5237:
5233:
5225:
5221:
5214:
5210:
5199:
5195:
5188:
5170:
5166:
5125:
5121:
5108:
5104:
5088:
5087:
5080:
5078:
5067:
5063:
5028:
5024:
5019:
4955:
4948:
4945:
4916:
4912:
4908:
4884:
4876:
4873:
4872:
4831:
4819:
4815:
4813:
4810:
4809:
4795:
4769:
4757:
4748:
4747:
4746:
4728:
4724:
4715:
4711:
4709:
4706:
4705:
4686:
4683:
4682:
4661:
4648:
4647:
4646:
4630:
4617:
4616:
4615:
4605:
4592:
4591:
4590:
4582:
4579:
4578:
4559:
4551:
4543:
4540:
4539:
4515:
4502:
4501:
4500:
4488:
4477:
4465:
4442:
4440:
4437:
4436:
4412:
4399:
4398:
4397:
4391:
4380:
4368:
4360:
4358:
4355:
4354:
4329:
4326:
4325:
4309:
4306:
4305:
4283:
4280:
4279:
4263:
4260:
4259:
4248:
4226:
4222:
4220:
4217:
4216:
4200:
4197:
4196:
4170:
4166:
4136:
4132:
4123:
4119:
4117:
4114:
4113:
4110:
4084:
4080:
4078:
4075:
4074:
4057:
4053:
4039:
4036:
4035:
4019:
4016:
4015:
3998:
3994:
3992:
3989:
3988:
3969:
3966:
3965:
3962:
3940:
3936:
3934:
3931:
3930:
3913:
3909:
3907:
3904:
3903:
3887:
3884:
3883:
3858:
3855:
3854:
3838:
3835:
3834:
3810:
3796:
3793:
3792:
3762:
3754:
3746:
3723:
3722:
3718:
3706:
3678:
3674:
3672:
3669:
3668:
3639:
3631:
3623:
3610:
3593:
3592:
3588:
3576:
3548:
3544:
3542:
3539:
3538:
3518:
3515:
3514:
3498:
3495:
3494:
3471:expander graphs
3463:
3457:
3445:discrete groups
3406:
3403:
3402:
3358:
3343:
3339:
3316:
3313:
3312:
3299:Euclidean plane
3285:be for a given
3260:
3256:
3254:
3251:
3250:
3189:
3168:
3164:
3162:
3159:
3158:
3108:
3104:
3097:
3095:
3080:
3058:
3054:
3052:
3049:
3048:
2957:
2954:
2953:
2938:
2897:
2894:
2893:
2869:
2865:
2861:
2855:
2851:
2832:
2816:
2812:
2777:
2774:
2773:
2737:
2732:
2731:
2729:
2726:
2725:
2719:
2694:
2689:
2688:
2673:
2669:
2661:
2658:
2657:
2638:
2627:
2619:
2614:
2613:
2612:
2608:
2598:
2591:
2586:
2573:
2569:
2556:
2540:
2513:
2509:
2504:
2503:
2495:
2487:
2482:
2481:
2480:
2476:
2475:
2471:
2470:
2468:
2465:
2464:
2444:
2439:
2438:
2436:
2433:
2432:
2388:
2383:
2382:
2380:
2377:
2376:
2369:
2326:
2321:
2315:
2312:
2311:
2273:
2268:
2251:
2235:
2231:
2220:
2219:
2210:
2206:
2196:
2192:
2187:
2177:
2174:
2173:
2146:
2141:
2140:
2132:
2129:
2128:
2114:Osserman (1978)
2091:
2088:
2087:
2071:
2068:
2067:
2039:
2036:
2035:
2019:
2016:
2015:
1998:
1994:
1982:
1978:
1976:
1973:
1972:
1956:
1953:
1952:
1936:
1933:
1932:
1876:
1873:
1872:
1826:
1823:
1822:
1806:
1803:
1802:
1786:
1783:
1782:
1781:if and only if
1766:
1763:
1762:
1747:smooth boundary
1721:
1716:
1715:
1713:
1710:
1709:
1693:
1690:
1689:
1668:
1663:
1662:
1653:
1649:
1647:
1644:
1643:
1616:
1612:
1608:
1602:
1598:
1578:
1562:
1558:
1523:
1520:
1519:
1488:
1485:
1484:
1453:
1450:
1449:
1429:
1424:
1423:
1415:
1412:
1411:
1404:
1377:
1373:
1367:
1363:
1361:
1340:
1336:
1334:
1331:
1330:
1265:
1261:
1259:
1256:
1255:
1221:
1196:
1193:
1192:
1176:
1173:
1172:
1152:
1149:
1148:
1132:
1129:
1128:
1111:
1100:
1099:
1098:
1084:
1081:
1080:
1053:
1042:
1041:
1040:
1036:
1009:
1005:
1003:
1000:
999:
976:
973:
972:
941:
925:
920:
911:
907:
905:
902:
901:
887:
877:
848:
844:
833:
831:
823:
820:
819:
802:Green's theorem
736:
732:
718:
715:
714:
692:
653:Johannes Kepler
601:
593:surface tension
510:
506:
504:
501:
500:
472:
467:
466:
464:
461:
460:
416:
413:
412:
392:
387:
386:
384:
381:
380:
379:is a sphere in
364:
361:
360:
339:
334:
333:
324:
320:
318:
315:
314:
289:
285:
281:
275:
271:
251:
235:
231:
196:
193:
192:
161:
158:
157:
137:
132:
131:
123:
120:
119:
91:
88:
87:
63:
58:
57:
55:
52:
51:
35:
32:
31:
17:
12:
11:
5:
6669:
6659:
6658:
6653:
6648:
6643:
6638:
6621:
6620:
6618:
6617:
6612:
6607:
6602:
6597:
6591:
6589:
6585:
6584:
6581:
6580:
6578:
6577:
6572:
6567:
6562:
6561:
6560:
6550:
6544:
6542:
6533:
6532:
6530:
6529:
6524:
6522:Sard's theorem
6519:
6514:
6509:
6508:
6507:
6505:Lifting theory
6496:
6490:
6486:
6485:
6483:
6482:
6477:
6472:
6467:
6462:
6461:
6460:
6458:FubiniâTonelli
6450:
6445:
6440:
6439:
6438:
6433:
6428:
6420:
6419:
6418:
6413:
6408:
6400:
6394:
6392:
6386:
6385:
6383:
6382:
6377:
6372:
6367:
6362:
6357:
6352:
6346:
6341:
6340:
6339:
6337:in probability
6334:
6324:
6319:
6314:
6308:
6307:
6306:
6301:
6296:
6285:
6283:
6279:
6278:
6276:
6275:
6270:
6265:
6260:
6255:
6250:
6249:
6248:
6238:
6233:
6232:
6231:
6221:
6216:
6211:
6206:
6201:
6196:
6191:
6186:
6180:
6178:
6172:
6171:
6169:
6168:
6163:
6158:
6153:
6148:
6143:
6138:
6133:
6128:
6123:
6118:
6117:
6116:
6111:
6106:
6096:
6091:
6086:
6081:
6071:
6066:
6061:
6056:
6051:
6046:
6044:Locally finite
6041:
6031:
6026:
6021:
6016:
6011:
6006:
5996:
5991:
5986:
5981:
5976:
5971:
5966:
5961:
5956:
5950:
5948:
5941:
5940:
5938:
5937:
5932:
5927:
5922:
5917:
5916:
5915:
5905:
5900:
5892:
5887:
5886:
5885:
5875:
5870:
5869:
5868:
5858:
5853:
5848:
5847:
5846:
5836:
5831:
5826:
5820:
5818:
5814:
5813:
5811:
5810:
5801:
5800:
5799:
5789:
5784:
5776:
5771:
5761:
5759:
5758:Basic concepts
5755:
5754:
5751:Measure theory
5747:
5746:
5739:
5732:
5724:
5718:
5717:
5708:
5703:
5692:
5691:External links
5689:
5687:
5686:
5656:
5631:
5603:(2): 726â739.
5592:
5564:
5555:
5543:(4): 439â561.
5539:. New Series.
5526:Widgerson, Avi
5522:Linial, Nathan
5517:
5508:Hadwiger, Hugo
5504:
5494:
5487:
5471:
5465:
5445:
5433:
5427:
5410:
5391:
5373:
5367:
5351:BollobĂĄs, BĂ©la
5347:
5325:
5323:
5320:
5317:
5316:
5300:
5287:
5275:
5263:
5256:Calabro (2004)
5247:
5231:
5219:
5208:
5193:
5186:
5164:
5135:(2): 726â739.
5119:
5102:
5061:
5042:(6): 526â566.
5021:
5020:
5018:
5015:
5014:
5013:
5008:
5003:
4998:
4993:
4988:
4983:
4981:Expander graph
4978:
4973:
4967:
4961:
4960:
4944:
4941:
4940:
4939:
4928:
4923:
4919:
4915:
4911:
4907:
4904:
4901:
4898:
4893:
4889:
4883:
4880:
4858:
4857:
4846:
4843:
4840:
4835:
4830:
4827:
4822:
4818:
4794:
4791:
4790:
4789:
4778:
4772:
4766:
4763:
4760:
4756:
4751:
4745:
4742:
4739:
4736:
4731:
4727:
4723:
4718:
4714:
4690:
4679:
4678:
4664:
4659:
4656:
4651:
4645:
4642:
4639:
4633:
4628:
4625:
4620:
4614:
4608:
4603:
4600:
4595:
4589:
4586:
4562:
4558:
4554:
4550:
4547:
4536:
4535:
4524:
4518:
4513:
4510:
4505:
4497:
4494:
4491:
4486:
4483:
4480:
4476:
4472:
4468:
4464:
4461:
4458:
4455:
4452:
4449:
4445:
4430:
4429:
4415:
4410:
4407:
4402:
4394:
4389:
4386:
4383:
4379:
4375:
4371:
4367:
4363:
4339:
4336:
4333:
4313:
4293:
4290:
4287:
4267:
4256:Hamming weight
4247:
4244:
4229:
4225:
4204:
4184:
4181:
4178:
4173:
4169:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4144:
4139:
4135:
4131:
4126:
4122:
4109:
4106:
4087:
4083:
4060:
4056:
4052:
4049:
4046:
4043:
4023:
4001:
3997:
3973:
3961:
3958:
3943:
3939:
3916:
3912:
3891:
3871:
3868:
3865:
3862:
3842:
3822:
3817:
3814:
3809:
3806:
3803:
3800:
3789:
3788:
3776:
3772:
3769:
3765:
3761:
3757:
3753:
3749:
3745:
3742:
3739:
3736:
3733:
3730:
3726:
3721:
3715:
3712:
3709:
3705:
3701:
3698:
3695:
3692:
3689:
3686:
3681:
3677:
3665:
3653:
3649:
3646:
3642:
3638:
3634:
3630:
3626:
3622:
3617:
3614:
3609:
3606:
3603:
3600:
3596:
3591:
3585:
3582:
3579:
3575:
3571:
3568:
3565:
3562:
3559:
3556:
3551:
3547:
3522:
3502:
3461:Expander graph
3459:Main article:
3456:
3453:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3397:is called the
3395:
3394:
3383:
3380:
3377:
3374:
3371:
3368:
3365:
3361:
3357:
3354:
3351:
3346:
3342:
3338:
3335:
3332:
3329:
3326:
3323:
3320:
3274:
3271:
3268:
3263:
3259:
3232:
3231:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3192:
3188:
3185:
3182:
3179:
3176:
3171:
3167:
3152:
3151:
3140:
3135:
3131:
3128:
3125:
3122:
3119:
3116:
3111:
3107:
3103:
3100:
3092:
3089:
3086:
3083:
3079:
3078:lim inf
3075:
3072:
3069:
3066:
3061:
3057:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2937:
2934:
2901:
2890:
2889:
2876:
2872:
2868:
2864:
2858:
2854:
2850:
2847:
2844:
2839:
2835:
2831:
2828:
2825:
2822:
2819:
2815:
2811:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2740:
2735:
2718:
2715:
2702:
2697:
2692:
2687:
2682:
2679:
2676:
2672:
2668:
2665:
2654:
2653:
2641:
2637:
2634:
2630:
2622:
2617:
2611:
2605:
2601:
2597:
2594:
2589:
2585:
2579:
2576:
2572:
2568:
2563:
2559:
2555:
2552:
2549:
2546:
2543:
2538:
2532:
2529:
2526:
2523:
2520:
2516:
2512:
2507:
2502:
2498:
2490:
2485:
2479:
2474:
2447:
2442:
2391:
2386:
2367:
2335:
2332:
2329:
2324:
2320:
2308:
2307:
2296:
2293:
2290:
2287:
2282:
2279:
2276:
2271:
2267:
2263:
2258:
2254:
2250:
2247:
2244:
2241:
2238:
2234:
2227:
2224:
2218:
2213:
2209:
2203:
2199:
2195:
2190:
2186:
2181:
2149:
2144:
2139:
2136:
2101:
2098:
2095:
2075:
2055:
2052:
2049:
2046:
2043:
2034:, subtracting
2023:
2001:
1997:
1993:
1990:
1985:
1981:
1960:
1940:
1931:between a set
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1810:
1790:
1770:
1759:Hadwiger (1957
1755:Baebler (1957)
1724:
1719:
1697:
1671:
1666:
1661:
1656:
1652:
1640:
1639:
1628:
1623:
1619:
1615:
1611:
1605:
1601:
1597:
1594:
1591:
1585:
1581:
1577:
1574:
1571:
1568:
1565:
1561:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1504:
1501:
1498:
1495:
1492:
1469:
1466:
1463:
1460:
1457:
1432:
1427:
1422:
1419:
1403:
1400:
1399:
1398:
1387:
1380:
1376:
1370:
1366:
1360:
1357:
1354:
1351:
1348:
1343:
1339:
1309:
1308:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1268:
1264:
1235:the length of
1220:
1217:
1200:
1180:
1169:Wigner caustic
1156:
1136:
1114:
1107:
1104:
1097:
1094:
1091:
1088:
1077:
1076:
1065:
1061:
1056:
1049:
1046:
1039:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1012:
1008:
980:
969:
968:
957:
951:
948:
944:
940:
937:
934:
931:
928:
924:
919:
914:
910:
867:
866:
851:
847:
842:
839:
836:
830:
827:
790:Fourier series
781:in this case.
760:area of a disk
756:
755:
744:
739:
735:
731:
728:
725:
722:
691:
688:
619:The classical
600:
597:
581:Ancient Greece
561:Dido's problem
542:
541:
530:
527:
524:
521:
518:
513:
509:
475:
470:
426:
423:
420:
395:
390:
368:
342:
337:
332:
327:
323:
311:
310:
296:
292:
288:
284:
278:
274:
270:
267:
264:
258:
254:
250:
247:
244:
241:
238:
234:
230:
227:
224:
221:
218:
215:
212:
209:
206:
203:
200:
177:
174:
171:
168:
165:
140:
135:
130:
127:
107:
104:
101:
98:
95:
66:
61:
39:
15:
9:
6:
4:
3:
2:
6668:
6657:
6654:
6652:
6649:
6647:
6644:
6642:
6639:
6637:
6634:
6633:
6631:
6616:
6613:
6611:
6610:Real analysis
6608:
6606:
6603:
6601:
6598:
6596:
6593:
6592:
6590:
6586:
6576:
6573:
6571:
6568:
6566:
6563:
6559:
6556:
6555:
6554:
6551:
6549:
6546:
6545:
6543:
6540:
6534:
6528:
6525:
6523:
6520:
6518:
6515:
6513:
6510:
6506:
6503:
6502:
6501:
6498:
6497:
6494:
6491:
6489:Other results
6487:
6481:
6478:
6476:
6475:RadonâNikodym
6473:
6471:
6468:
6466:
6463:
6459:
6456:
6455:
6454:
6451:
6449:
6448:Fatou's lemma
6446:
6444:
6441:
6437:
6434:
6432:
6429:
6427:
6424:
6423:
6421:
6417:
6414:
6412:
6409:
6407:
6404:
6403:
6401:
6399:
6396:
6395:
6393:
6391:
6387:
6381:
6378:
6376:
6373:
6371:
6368:
6366:
6363:
6361:
6358:
6356:
6353:
6351:
6347:
6345:
6342:
6338:
6335:
6333:
6330:
6329:
6328:
6325:
6323:
6320:
6318:
6315:
6313:
6310:Convergence:
6309:
6305:
6302:
6300:
6297:
6295:
6292:
6291:
6290:
6287:
6286:
6284:
6280:
6274:
6271:
6269:
6266:
6264:
6261:
6259:
6256:
6254:
6251:
6247:
6244:
6243:
6242:
6239:
6237:
6234:
6230:
6227:
6226:
6225:
6222:
6220:
6217:
6215:
6212:
6210:
6207:
6205:
6202:
6200:
6197:
6195:
6192:
6190:
6187:
6185:
6182:
6181:
6179:
6177:
6173:
6167:
6164:
6162:
6159:
6157:
6154:
6152:
6149:
6147:
6144:
6142:
6139:
6137:
6134:
6132:
6129:
6127:
6124:
6122:
6119:
6115:
6114:Outer regular
6112:
6110:
6109:Inner regular
6107:
6105:
6104:Borel regular
6102:
6101:
6100:
6097:
6095:
6092:
6090:
6087:
6085:
6082:
6080:
6076:
6072:
6070:
6067:
6065:
6062:
6060:
6057:
6055:
6052:
6050:
6047:
6045:
6042:
6040:
6036:
6032:
6030:
6027:
6025:
6022:
6020:
6017:
6015:
6012:
6010:
6007:
6005:
6001:
5997:
5995:
5992:
5990:
5987:
5985:
5982:
5980:
5977:
5975:
5972:
5970:
5967:
5965:
5962:
5960:
5957:
5955:
5952:
5951:
5949:
5947:
5942:
5936:
5933:
5931:
5928:
5926:
5923:
5921:
5918:
5914:
5911:
5910:
5909:
5906:
5904:
5901:
5899:
5893:
5891:
5888:
5884:
5881:
5880:
5879:
5876:
5874:
5871:
5867:
5864:
5863:
5862:
5859:
5857:
5854:
5852:
5849:
5845:
5842:
5841:
5840:
5837:
5835:
5832:
5830:
5827:
5825:
5822:
5821:
5819:
5815:
5809:
5805:
5802:
5798:
5795:
5794:
5793:
5792:Measure space
5790:
5788:
5785:
5783:
5781:
5777:
5775:
5772:
5770:
5766:
5763:
5762:
5760:
5756:
5752:
5745:
5740:
5738:
5733:
5731:
5726:
5725:
5722:
5716:
5712:
5709:
5707:
5704:
5702:
5698:
5695:
5694:
5682:
5678:
5674:
5670:
5666:
5662:
5657:
5652:
5648:
5644:
5640:
5636:
5632:
5628:
5624:
5620:
5616:
5611:
5606:
5602:
5598:
5593:
5587:
5582:
5578:
5574:
5570:
5565:
5561:
5556:
5551:
5546:
5542:
5538:
5531:
5527:
5523:
5518:
5513:
5509:
5505:
5502:
5498:
5495:
5490:
5488:3-540-60656-4
5484:
5480:
5476:
5472:
5468:
5466:9780914351023
5462:
5457:
5456:
5450:
5446:
5442:
5438:
5434:
5430:
5424:
5420:
5416:
5411:
5397:
5392:
5389:
5385:
5384:
5379:
5374:
5370:
5364:
5359:
5358:
5352:
5348:
5346:
5345:0-387-05889-3
5342:
5338:
5334:
5330:
5327:
5326:
5314:
5310:
5304:
5297:
5291:
5285:
5279:
5273:
5272:Leader (1991)
5267:
5261:
5257:
5251:
5245:
5241:
5235:
5229:
5223:
5217:
5212:
5206:
5202:
5197:
5189:
5187:9780817645830
5183:
5179:
5175:
5168:
5160:
5156:
5152:
5148:
5143:
5138:
5134:
5130:
5123:
5116:
5112:
5106:
5098:
5092:
5076:
5072:
5065:
5057:
5053:
5049:
5045:
5041:
5037:
5033:
5026:
5022:
5012:
5009:
5007:
5004:
5002:
4999:
4997:
4994:
4992:
4989:
4987:
4984:
4982:
4979:
4977:
4974:
4971:
4968:
4966:
4963:
4962:
4958:
4952:
4947:
4926:
4921:
4917:
4913:
4905:
4902:
4899:
4891:
4887:
4881:
4878:
4871:
4870:
4869:
4867:
4863:
4844:
4841:
4838:
4833:
4828:
4825:
4820:
4816:
4808:
4807:
4806:
4804:
4800:
4776:
4764:
4761:
4758:
4754:
4743:
4737:
4734:
4729:
4725:
4716:
4704:
4703:
4702:
4688:
4657:
4654:
4643:
4640:
4637:
4626:
4623:
4612:
4601:
4598:
4587:
4584:
4577:
4576:
4575:
4556:
4548:
4545:
4522:
4511:
4508:
4495:
4492:
4489:
4484:
4481:
4478:
4474:
4470:
4459:
4450:
4447:
4435:
4434:
4433:
4408:
4405:
4392:
4387:
4384:
4381:
4377:
4373:
4365:
4353:
4352:
4351:
4337:
4334:
4331:
4311:
4291:
4288:
4285:
4265:
4257:
4253:
4252:Hamming balls
4243:
4227:
4223:
4202:
4179:
4176:
4171:
4167:
4163:
4160:
4154:
4151:
4145:
4142:
4137:
4133:
4124:
4105:
4103:
4085:
4081:
4058:
4050:
4047:
4044:
4021:
3999:
3995:
3987:
3984:-dimensional
3971:
3957:
3941:
3914:
3889:
3866:
3840:
3812:
3807:
3804:
3798:
3774:
3770:
3767:
3759:
3751:
3743:
3734:
3719:
3713:
3710:
3707:
3699:
3693:
3690:
3687:
3679:
3666:
3651:
3647:
3644:
3636:
3628:
3612:
3607:
3604:
3598:
3589:
3583:
3580:
3577:
3569:
3563:
3560:
3557:
3549:
3536:
3535:
3534:
3520:
3513:and a number
3500:
3490:
3488:
3484:
3480:
3476:
3475:sparse graphs
3472:
3468:
3462:
3452:
3450:
3446:
3442:
3441:Cayley graphs
3423:
3420:
3417:
3414:
3411:
3400:
3378:
3375:
3369:
3363:
3352:
3344:
3340:
3330:
3324:
3318:
3311:
3310:
3309:
3308:The function
3306:
3304:
3300:
3296:
3292:
3288:
3269:
3261:
3257:
3248:
3243:
3241:
3237:
3215:
3212:
3206:
3203:
3200:
3194:
3186:
3183:
3180:
3174:
3169:
3165:
3157:
3156:
3155:
3138:
3133:
3126:
3120:
3117:
3109:
3105:
3098:
3090:
3087:
3081:
3073:
3067:
3059:
3055:
3047:
3046:
3045:
3044:
3040:
3036:
3032:
3028:
3024:
3020:
3016:
3015:Borel measure
3012:
3008:
3005:
3001:
2997:
2993:
2974:
2971:
2968:
2965:
2962:
2951:
2947:
2943:
2933:
2931:
2928:in 1992, and
2927:
2926:Bruce Kleiner
2923:
2919:
2915:
2899:
2874:
2870:
2866:
2856:
2852:
2845:
2842:
2837:
2833:
2826:
2823:
2820:
2809:
2803:
2800:
2797:
2794:
2788:
2782:
2779:
2772:
2771:
2770:
2768:
2764:
2760:
2756:
2755:Thierry Aubin
2738:
2723:
2714:
2695:
2680:
2677:
2674:
2670:
2666:
2663:
2635:
2620:
2609:
2603:
2599:
2595:
2592:
2587:
2583:
2577:
2574:
2570:
2566:
2561:
2557:
2550:
2547:
2544:
2536:
2527:
2524:
2521:
2514:
2510:
2500:
2488:
2477:
2472:
2463:
2462:
2461:
2445:
2430:
2426:
2421:
2419:
2415:
2411:
2407:
2389:
2374:
2370:
2363:
2359:
2355:
2351:
2333:
2330:
2327:
2322:
2318:
2291:
2280:
2277:
2274:
2269:
2265:
2261:
2256:
2252:
2245:
2242:
2239:
2222:
2211:
2207:
2201:
2197:
2193:
2188:
2184:
2179:
2172:
2171:
2170:
2169:
2165:
2147:
2137:
2134:
2126:
2121:
2120:, §3.2.43)).
2119:
2118:Federer (1969
2115:
2099:
2093:
2073:
2050:
2044:
2041:
2021:
1999:
1995:
1991:
1988:
1983:
1979:
1958:
1938:
1930:
1925:
1911:
1905:
1899:
1896:
1893:
1887:
1881:
1878:
1855:
1849:
1846:
1843:
1837:
1831:
1828:
1808:
1788:
1768:
1760:
1756:
1752:
1751:Schmidt (1949
1748:
1744:
1740:
1722:
1708:is a ball in
1695:
1687:
1669:
1659:
1654:
1650:
1626:
1621:
1617:
1613:
1603:
1599:
1592:
1589:
1583:
1579:
1572:
1569:
1566:
1555:
1549:
1546:
1543:
1540:
1534:
1528:
1525:
1518:
1517:
1516:
1499:
1493:
1490:
1483:
1464:
1458:
1455:
1448:
1430:
1420:
1417:
1409:
1385:
1378:
1374:
1368:
1364:
1358:
1355:
1352:
1349:
1346:
1341:
1337:
1329:
1328:
1327:
1325:
1320:
1318:
1313:
1295:
1289:
1286:
1283:
1280:
1274:
1271:
1266:
1262:
1254:
1253:
1252:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1216:
1214:
1198:
1178:
1170:
1154:
1134:
1112:
1105:
1102:
1095:
1092:
1089:
1086:
1063:
1059:
1054:
1047:
1044:
1037:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1010:
1006:
998:
997:
996:
994:
978:
955:
946:
942:
938:
932:
929:
926:
922:
917:
912:
908:
900:
899:
898:
896:
891:
886:is at least 4
884:
880:
876:
872:
849:
845:
840:
837:
834:
828:
825:
818:
817:
816:
814:
809:
807:
803:
799:
795:
791:
787:
782:
780:
777:
773:
769:
765:
761:
742:
737:
733:
729:
726:
723:
720:
713:
712:
711:
709:
705:
701:
697:
687:
684:
679:
677:
676:
671:
670:Jakob Steiner
666:
664:
660:
659:
654:
650:
646:
643:, considered
642:
638:
634:
629:
626:
622:
613:
605:
596:
594:
589:
584:
582:
578:
574:
570:
566:
562:
558:
554:
550:
545:
528:
525:
522:
519:
516:
511:
507:
499:
498:
497:
495:
491:
473:
458:
454:
453:
452:Isoperimetric
448:
444:
440:
439:circumference
424:
421:
418:
409:
393:
366:
358:
340:
330:
325:
321:
294:
290:
286:
276:
272:
265:
262:
256:
252:
245:
242:
239:
228:
222:
219:
216:
213:
207:
201:
198:
191:
190:
189:
172:
166:
163:
156:
138:
128:
125:
102:
96:
93:
86:
82:
64:
37:
29:
26:
22:
6547:
6390:Main results
6126:Set function
6054:Metric outer
6009:Decomposable
5866:Cylinder set
5779:
5715:cut-the-knot
5664:
5660:
5642:
5638:
5600:
5596:
5576:
5572:
5559:
5540:
5536:
5511:
5500:
5478:
5454:
5440:
5414:
5402:. Retrieved
5381:
5356:
5332:
5308:
5303:
5295:
5290:
5278:
5266:
5250:
5234:
5222:
5211:
5196:
5177:
5167:
5132:
5128:
5122:
5114:
5110:
5109:J. Steiner,
5105:
5079:. Retrieved
5077:(in Spanish)
5074:
5064:
5039:
5035:
5025:
5011:Mixed volume
4859:
4805:states that
4802:
4798:
4796:
4680:
4574:of the form
4537:
4431:
4251:
4249:
4111:
3963:
3790:
3491:
3473:, which are
3467:graph theory
3464:
3448:
3398:
3396:
3307:
3294:
3290:
3286:
3246:
3244:
3239:
3235:
3233:
3153:
3038:
3034:
3022:
3018:
3010:
3006:
3000:metric space
2995:
2991:
2949:
2939:
2891:
2759:Misha Gromov
2720:
2655:
2424:
2422:
2413:
2405:
2365:
2361:
2357:
2349:
2309:
2125:Federer 1969
2122:
1926:
1641:
1447:surface area
1405:
1323:
1321:
1314:
1310:
1251:states that
1248:
1244:
1240:
1236:
1232:
1224:
1222:
1078:
992:
970:
894:
892:
882:
878:
870:
868:
812:
810:
783:
778:
775:
771:
767:
763:
757:
710:states that
707:
703:
699:
693:
680:
674:
667:
662:
656:
630:
620:
618:
585:
560:
553:plane figure
548:
546:
543:
493:
489:
450:
443:closed curve
410:
312:
81:surface area
20:
18:
6350:compact set
6317:of measures
6253:Pushforward
6246:Projections
6236:Logarithmic
6079:Probability
6069:Pre-measure
5851:Borel space
5769:of measures
5701:Convergence
5639:Math. Nachr
2930:Chris Croke
2763:Yuri Burago
2410:rectifiable
2166:has finite
1219:On a sphere
357:unit sphere
6630:Categories
6322:in measure
6049:Maximising
6019:Equivalent
5913:Vitali set
5610:1512.06684
5497:Gromov, M.
5404:8 February
5322:References
5142:1512.06684
5081:14 January
4432:satisfies
3455:For graphs
3031:measurable
2918:André Weil
1821:such that
1743:regularity
804:, and the
798:arc length
762:of radius
696:inequality
690:On a plane
645:rotational
28:inequality
6436:Maharam's
6406:Dominated
6219:Intensity
6214:Hausdorff
6121:Saturated
6039:Invariant
5944:Types of
5903:Ï-algebra
5873:đ-system
5839:Borel set
5834:Baire set
5681:123704157
5667:: 52â65.
5627:119708226
5388:EMS Press
5159:119708226
4882:≤
4839:⋅
4826:≥
4801:and area
4713:Φ
4641:⋯
4475:∑
4471:≥
4454:Γ
4451:∪
4378:∑
4374:≥
4335:⊆
4177:
4164:−
4152:≥
4121:Φ
3986:hypercube
3938:Φ
3911:Φ
3861:Γ
3816:¯
3741:∖
3729:Γ
3711:⊆
3676:Φ
3616:¯
3581:⊆
3546:Φ
3418:μ
3364:μ
3341:μ
3258:μ
3236:extension
3234:is the Δ-
3216:ε
3213:≤
3184:∈
3170:ε
3134:ε
3121:μ
3118:−
3110:ε
3099:μ
3085:→
3082:ε
3056:μ
2969:μ
2846:
2824:−
2804:
2795:≥
2783:
2667:∈
2633:∇
2610:∫
2593:−
2584:ω
2575:−
2567:≤
2548:−
2525:−
2478:∫
2373:unit ball
2331:−
2323:∗
2289:∂
2278:−
2270:∗
2262:≤
2243:−
2226:¯
2185:ω
2138:⊂
2097:→
2094:ϵ
2074:ϵ
2045:
1992:ϵ
1984:ϵ
1959:ϵ
1900:
1882:
1850:
1832:
1739:convexity
1686:unit ball
1660:⊂
1593:
1570:−
1550:
1541:≥
1529:
1494:
1459:
1421:⊂
1359:−
1353:π
1347:≥
1317:Paul LĂ©vy
1287:−
1284:π
1272:≥
1106:~
1048:~
1034:π
1022:π
1016:⩾
939:π
933:
923:π
838:π
730:≤
724:π
665:, 1596).
523:π
517:≥
457:perimeter
331:⊂
266:
243:−
223:
214:≥
202:
167:
129:⊂
118:of a set
97:
85:perimeter
25:geometric
6453:Fubini's
6443:Egorov's
6411:Monotone
6370:variable
6348:Random:
6299:Strongly
6224:Lebesgue
6209:Harmonic
6199:Gaussian
6184:Counting
6151:Spectral
6146:Singular
6136:s-finite
6131:Ï-finite
6014:Discrete
5989:Complete
5946:Measures
5920:Null set
5808:function
5528:(2006).
5510:(1957).
5477:(1969).
5353:(1986).
5329:Blaschke
5091:cite web
5056:30037526
4943:See also
4258:at most
2922:Hadamard
2656:for all
2348:is the (
897:-gon is
588:surfaces
573:Carthage
557:boundary
445:and the
6365:process
6360:measure
6355:element
6294:Bochner
6268:Trivial
6263:Tangent
6241:Product
6099:Regular
6077:)
6064:Perfect
6037:)
6002:)
5994:Content
5984:Complex
5925:Support
5898:-system
5787:Measure
5075:El PaĂs
3297:is the
3043:lim inf
3033:subset
3029:, of a
2360:is the
2164:closure
1971:, i.e.
1239:and by
786:Hurwitz
637:physics
153:by its
6431:Jordan
6416:Vitali
6375:vector
6304:Weakly
6166:Vector
6141:Signed
6094:Random
6035:Quasi-
6024:Finite
6004:Convex
5964:Banach
5954:Atomic
5782:spaces
5767:
5679:
5625:
5485:
5463:
5425:
5365:
5343:
5184:
5157:
5054:
3293:). If
3154:where
3021:. The
3009:, and
3004:metric
2952:. Let
2765:, and
2310:where
2162:whose
1642:where
1482:volume
1408:sphere
1247:. The
1229:sphere
1079:where
683:convex
649:circle
625:curves
577:circle
313:where
155:volume
6273:Young
6194:Euler
6189:Dirac
6161:Tight
6089:Radon
6059:Outer
6029:Inner
5979:Brown
5974:Borel
5969:Besov
5959:Baire
5677:S2CID
5623:S2CID
5605:arXiv
5533:(PDF)
5399:(PDF)
5155:S2CID
5137:arXiv
5052:JSTOR
5017:Notes
4350:with
3791:Here
3025:, or
3013:is a
3002:with
2998:is a
2990:be a
1684:is a
1445:with
1211:is a
441:of a
355:is a
23:is a
6537:For
6426:Hahn
6282:Maps
6204:Haar
6075:Sub-
5829:Atom
5817:Sets
5483:ISBN
5461:ISBN
5423:ISBN
5406:2011
5363:ISBN
5341:ISBN
5270:cf.
5254:Cf.
5238:See
5182:ISBN
5097:link
5083:2021
3964:The
3929:and
3853:and
2423:The
1871:and
1480:and
1223:Let
971:Let
569:Dido
547:The
447:area
5713:at
5699:at
5669:doi
5647:doi
5615:doi
5601:442
5581:doi
5545:doi
5258:or
5147:doi
5133:442
5044:doi
5040:112
4168:log
3704:min
3574:min
3465:In
3443:of
3334:inf
3238:of
3037:of
3017:on
2843:vol
2801:vol
2780:per
2431:on
2408:is
2375:in
2042:vol
1897:per
1879:per
1847:vol
1829:vol
1590:vol
1547:vol
1526:per
1491:vol
1456:per
1171:of
1113:0.5
1055:0.5
930:tan
766:is
635:in
565:arc
263:vol
220:vol
199:per
164:vol
94:per
83:or
6632::
5675:.
5663:.
5641:.
5621:.
5613:.
5599:.
5577:84
5575:.
5571:.
5541:43
5535:.
5524:;
5421:.
5417:.
5386:,
5380:,
5176:.
5153:.
5145:.
5131:.
5115:18
5093:}}
5089:{{
5073:.
5050:.
5038:.
5034:.
4829:12
4242:.
3489:.
3242:.
2994::
2761:,
2757:,
2713:.
2420:.
2356:,
2116:;
2100:0.
1745:,
1741:,
1215:.
808:.
772:ÏR
768:ÏR
408:.
188:,
6073:(
6033:(
5998:(
5896:Ï
5806:/
5780:L
5743:e
5736:t
5729:v
5685:.
5683:.
5671::
5665:8
5655:.
5653:.
5649::
5643:2
5629:.
5617::
5607::
5591:.
5589:.
5583::
5553:.
5547::
5516:.
5493:.
5491:.
5469:.
5431:.
5408:.
5371:.
5190:.
5161:.
5149::
5139::
5099:)
5085:.
5058:.
5046::
4927:.
4922:3
4918:/
4914:2
4910:)
4906:c
4903:b
4900:a
4897:(
4892:4
4888:3
4879:T
4845:,
4842:T
4834:3
4821:2
4817:p
4803:T
4799:p
4777:.
4771:)
4765:1
4762:+
4759:r
4755:d
4750:(
4744:=
4741:)
4738:k
4735:,
4730:d
4726:Q
4722:(
4717:V
4689:r
4663:)
4658:r
4655:d
4650:(
4644:+
4638:+
4632:)
4627:1
4624:d
4619:(
4613:+
4607:)
4602:0
4599:d
4594:(
4588:=
4585:k
4561:|
4557:S
4553:|
4549:=
4546:k
4523:.
4517:)
4512:i
4509:d
4504:(
4496:1
4493:+
4490:r
4485:0
4482:=
4479:i
4467:|
4463:)
4460:S
4457:(
4448:S
4444:|
4414:)
4409:i
4406:d
4401:(
4393:r
4388:0
4385:=
4382:i
4370:|
4366:S
4362:|
4338:V
4332:S
4312:r
4292:1
4289:+
4286:r
4266:r
4228:d
4224:Q
4203:S
4183:)
4180:k
4172:2
4161:d
4158:(
4155:k
4149:)
4146:k
4143:,
4138:d
4134:Q
4130:(
4125:E
4086:d
4082:Q
4059:d
4055:}
4051:1
4048:,
4045:0
4042:{
4022:d
4000:d
3996:Q
3972:d
3942:V
3915:E
3890:S
3870:)
3867:S
3864:(
3841:S
3821:)
3813:S
3808:,
3805:S
3802:(
3799:E
3775:}
3771:k
3768:=
3764:|
3760:S
3756:|
3752::
3748:|
3744:S
3738:)
3735:S
3732:(
3725:|
3720:{
3714:V
3708:S
3700:=
3697:)
3694:k
3691:,
3688:G
3685:(
3680:V
3652:}
3648:k
3645:=
3641:|
3637:S
3633:|
3629::
3625:|
3621:)
3613:S
3608:,
3605:S
3602:(
3599:E
3595:|
3590:{
3584:V
3578:S
3570:=
3567:)
3564:k
3561:,
3558:G
3555:(
3550:E
3521:k
3501:G
3449:A
3427:)
3424:d
3421:,
3415:,
3412:X
3409:(
3382:}
3379:a
3376:=
3373:)
3370:A
3367:(
3360:|
3356:)
3353:A
3350:(
3345:+
3337:{
3331:=
3328:)
3325:a
3322:(
3319:I
3295:X
3291:A
3289:(
3287:Ό
3273:)
3270:A
3267:(
3262:+
3247:X
3240:A
3219:}
3210:)
3207:A
3204:,
3201:x
3198:(
3195:d
3191:|
3187:X
3181:x
3178:{
3175:=
3166:A
3139:,
3130:)
3127:A
3124:(
3115:)
3106:A
3102:(
3091:+
3088:0
3074:=
3071:)
3068:A
3065:(
3060:+
3039:X
3035:A
3019:X
3011:Ό
3007:d
2996:X
2978:)
2975:d
2972:,
2966:,
2963:X
2960:(
2900:S
2875:n
2871:/
2867:1
2863:)
2857:1
2853:B
2849:(
2838:n
2834:/
2830:)
2827:1
2821:n
2818:(
2814:)
2810:S
2807:(
2798:n
2792:)
2789:S
2786:(
2739:n
2734:R
2701:)
2696:n
2691:R
2686:(
2681:1
2678:,
2675:1
2671:W
2664:u
2640:|
2636:u
2629:|
2621:n
2616:R
2604:n
2600:/
2596:1
2588:n
2578:1
2571:n
2562:n
2558:/
2554:)
2551:1
2545:n
2542:(
2537:)
2531:)
2528:1
2522:n
2519:(
2515:/
2511:n
2506:|
2501:u
2497:|
2489:n
2484:R
2473:(
2446:n
2441:R
2425:n
2414:n
2406:S
2390:n
2385:R
2368:n
2366:Ï
2362:n
2358:L
2350:n
2334:1
2328:n
2319:M
2295:)
2292:S
2286:(
2281:1
2275:n
2266:M
2257:n
2253:/
2249:)
2246:1
2240:n
2237:(
2233:)
2223:S
2217:(
2212:n
2208:L
2202:n
2198:/
2194:1
2189:n
2180:n
2148:n
2143:R
2135:S
2112:(
2054:)
2051:S
2048:(
2022:n
2000:1
1996:B
1989:=
1980:B
1939:S
1912:.
1909:)
1906:S
1903:(
1894:=
1891:)
1888:B
1885:(
1859:)
1856:S
1853:(
1844:=
1841:)
1838:B
1835:(
1809:B
1789:S
1769:S
1723:n
1718:R
1696:S
1670:n
1665:R
1655:1
1651:B
1627:,
1622:n
1618:/
1614:1
1610:)
1604:1
1600:B
1596:(
1584:n
1580:/
1576:)
1573:1
1567:n
1564:(
1560:)
1556:S
1553:(
1544:n
1538:)
1535:S
1532:(
1503:)
1500:S
1497:(
1468:)
1465:S
1462:(
1431:n
1426:R
1418:S
1386:.
1379:2
1375:R
1369:2
1365:A
1356:A
1350:4
1342:2
1338:L
1324:R
1296:,
1293:)
1290:A
1281:4
1278:(
1275:A
1267:2
1263:L
1245:C
1241:A
1237:C
1233:L
1225:C
1199:C
1179:C
1155:C
1135:C
1103:A
1096:,
1093:A
1090:,
1087:L
1064:,
1060:|
1045:A
1038:|
1031:8
1028:+
1025:A
1019:4
1011:2
1007:L
979:C
956:.
950:)
947:n
943:/
936:(
927:n
918:=
913:n
909:Q
895:n
888:Ï
883:A
881:/
879:L
871:Q
850:2
846:L
841:A
835:4
829:=
826:Q
779:R
776:Ï
764:R
743:,
738:2
734:L
727:A
721:4
704:A
700:L
661:(
529:,
526:A
520:4
512:2
508:L
494:A
490:L
474:2
469:R
425:2
422:=
419:n
394:n
389:R
367:S
341:n
336:R
326:1
322:B
309:,
295:n
291:/
287:1
283:)
277:1
273:B
269:(
257:n
253:/
249:)
246:1
240:n
237:(
233:)
229:S
226:(
217:n
211:)
208:S
205:(
176:)
173:S
170:(
139:n
134:R
126:S
106:)
103:S
100:(
65:n
60:R
38:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
