共查询到20条相似文献,搜索用时 15 毫秒
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We prove that for a measurable subset of S
n–1 with fixed Haar measure, the volume of its convex hull is minimized for a cap (i.e. a ball with respect to the geodesic measure). We solve a similar problem for symmetric sets and n=2, 3. As a consequence, we deduce a result concerning Gaussian measures of dilatations of convex, symmetric sets in R
2 and R
3.Partially supported by KBN (Poland), Grant No. 2 1094 91 01. 相似文献
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We define the set Kq,e ⊂ K of quasiconvex extreme points for compact sets K ⊂ MN×n and study its properties. We show that Kq,e is the smallest generator of Q(K)-the quasiconvex hull of K, in the sense that Q(Kq,e) = Q(K), and that for every compact subset W ⊂ Q(K) with Q(W) = Q(K), Kq,e ⊂ W. The set of quasiconvex extreme points relies on K only in the sense that . We also establish that Ke ⊂ Kq,e, where Ke is the set of extreme points of C(K)-the convex hull of K. We give various examples to show that Kq,e is not necessarily closed even when Q(K) is not convex; and that for some nonconvex Q(K), Kq,e = Ke. We apply the results to the two well and three well problems studied in martensitic phase transitions. 相似文献
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Luís Bandeira Pablo Pedregal 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2009,26(5):1621-1634
We introduce a method to find, in a systematic way, rank-one convex polynomials. We show how it works in several examples. It can also be applied to convexity along general cones. 相似文献
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We consider a centered Gaussian random field X = {X t : t ∈ T} with values in a Banach space $\mathbb{B}$ defined on a parametric set T equal to ? m or ? m . It is supposed that the distribution of X t is independent of t. We consider the asymptotic behavior of closed convex hulls W n = conv{X t : t ∈ T n}, where (T n ) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b n ) n≥1 with probability 1, (in the sense of Hausdorff distance), where the limit set is the concentration ellipsoid of . The asymptotic behavior of the mathematical expectations Ef(W n ), where f is some function, is also studied. 相似文献
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A spectral characterization is obtained for those normal operators which belong to the convex hull of the unitary orbit of a given normal operator on a finite-dimensional space. This is used to prove the following: if A and B are normal operators on an n-dimensional complex Hilbert space H with eigenvalues given by α1,…,αn and β1,…, βn respectively, and if A ? B is also normal, then for any unitarily invariant norm on L(H). 相似文献
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Fuchang Gao 《Israel Journal of Mathematics》2001,123(1):359-364
LetT be a precompact subset of a Hilbert space. The metric entropy of the convex hull ofT is estimated in terms of the metric entropy ofT, when the latter is of order εℒ2. The estimate is best possible. Thus, it answers a question left open in [CKP]. 相似文献
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If a setX ⊂E
n has non-emptyk-dimensional interior, or if some point isk-dimensional surrounded, then the classic theorem of E. Steinitz may be extended. For example ifX ⊂E
n has int
k
X ≠ 0, (0 ≦k≦n) and ifp ɛ int conX, thenp ɛ int conY for someY ⊂X with cardY≦2n−k+1. 相似文献
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John Wermer 《Arkiv f?r Matematik》1982,20(1-2):129-135
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Piet Groeneboom 《Probability Theory and Related Fields》1988,79(3):327-368
It is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally, after rescaling, to a strongly mixing Markov process, as the sample size tends to infinity. The structure of the limiting Markov process is determined explicitly, and from this a central limit theorem for the number of vertices of the convex hull is derived. Similar results are given for uniform samples from the unit disk. 相似文献
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Let be the graph of a continuous map of the unit sphere of into , and the polynomially convex hull of . Several examples of for are given, which have different properties from the known ones for .
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Theprofile of a hypergraph onn vertices is (f
0, f1, ...,f
n) wheref
i denotes the number ofi-element edges. The extreme points of the set of profiles is determined for certain hypergraph classes. The results contain
many old theorems of extremal set theory as particular cases (Sperner. Erdős—Ko—Rado, Daykin—Frankl—Green—Hilton). 相似文献
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José Rodríguez 《Mathematische Nachrichten》2007,280(11):1302-1309
Let (Ω, Σ, μ) be a complete probability space and let X be a Banach space. We consider the following problem: Given a function f: Ω → X for which there is a norming set B ? BX * such that Zf,B = {x * ○ f: x * ∈ B } is uniformly integrable and has the Bourgain property, does it follow that f is Birkhoff integrable? It turns out that this question is equivalent to the following one: Given a pointwise bounded family ?? ? ?Ω with the Bourgain property, does its convex hull co(??) have the Bourgain property? With the help of an example of D. H. Fremlin, we make clear that both questions have negative answer in general. We prove that a function f: Ω → X is scalarly measurable provided that there is a norming set B ? BX * such that Zf,B has the Bourgain property. As an application we show that the first problem has positive solution in several cases, for instance: (i) when BX * is weak* separable; (ii) under Martin's axiom, for functions defined on [0, 1] with values in a Banach space with density character smaller than the continuum. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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E. Michael and I. Namioka proved the following theorem. Let Y be a convex G δ -subset of a Banach space E such that if K ? Y is a compact space, then its closed (in Y) convex hull is also compact. Then every lower semicontinuous set-valued mapping of a paracompact space X to Y with closed (in Y) convex values has a continuous selection. E. Michael asked the question: Is the assumption that Y is G δ essential? In this note we give an affirmative answer to this question of Michael. 相似文献