Convex bodies, economic cap coverings, random polytopes

Let K be a d-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by K n the convex hull of n points chosen randomly and independently from K according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the sth intrinsic volumes V s (K n ) of K n for s ∈ {1, . . . , d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of K n . The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

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“…For more details and for further references on the economical cap covering theorem, see [3] and [6].…”

Section: An Immediate Consequence Of This Theorem Is That Mt λ D (K(t))mentioning

confidence: 99%

Intrinsic volumes of inscribed random polytopes in smooth convex bodies

2010

Self Cite

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“…Hence, the theorem implies that m ≈ 1 ε vol(K(ε)). Moreover, if K n is the random polytope on n points from K then K − K( 1 n ) is a good approximation to K n [4]; of course, this is why the study of K(ε) and, consequently, of cap coverings is of use here. It follows that, for m corresponding to ε = 1/n, the expected number of…”

Section: Lemma 14 Suppose F Is the Normal Density Or The Uniform Denmentioning

confidence: 99%

“…We give such a proof of Lemma 13 at the end of this section (the proof of Lemma 14 is similar). For a cube, however, things are more complicated and we will use an economic cap-covering [4]. The next theorem is from [3].…”

Section: Lemma 14 Suppose F Is the Normal Density Or The Uniform Denmentioning

confidence: 99%

Nash equilibria in random games

Bárány

Vempala

Vetta

2007

Random Struct Algorithms

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ABSTRACT:We consider Nash equilibria in 2-player random games and analyze a simple Las Vegas algorithm for finding an equilibrium. The algorithm is combinatorial and always finds a Nash equilibrium; on m × n payoff matrices, it runs in time O(m 2 n log log n + n 2 m log log m) with high probability. Our result follows from showing that a 2-player random game has a Nash equilibrium with supports of size two with high probability, at least 1 − O(1/ log n). Our main tool is a polytope formulation of equilibria.

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“…The floating body is the intersection of halfspaces, so it is convex. The general behaviour of E(K, n) was described in [10]: E(K, n) is of the same order of magnitude as the volume of the wet part with t = 1/n. This works for general convex bodies K ∈ K, not only when K is smooth or is a polytope.…”

Section: Minimal Caps and A General Resultsmentioning

confidence: 99%

“…We call K(v ≥ t) the floating body of K with parameter t > 0 as, in a similar picture, this is the part of K that floats above water (cf. [10] and [32]). The floating body is the intersection of halfspaces, so it is convex.…”

Section: Minimal Caps and A General Resultsmentioning

confidence: 99%

Random Polytopes, Convex Bodies, and Approximation

Weil¹

Stochastic Geometry

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Assume K ⊂ R d is a convex body and X n ⊂ K is a random sample of n uniform, independent points from K. The convex hull of X n is a convex polytope K n called random polytope inscribed in K. We are going to investigate various properties of this polytope: for instance how well it approximates K, or how many vertices and facets it has. It turns out that K n is very close to the so called floating body inscribed in K with parameter 1/n. To show this we develop and use the technique of cap coverings and Macbeath regions. Its power will be illustrated, besides random polytopes, on several examples: floating bodies, lattice polytopes, and approximation problems.

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Convex bodies, economic cap coverings, random polytopes

Cited by 149 publications

References 21 publications

Intrinsic volumes of inscribed random polytopes in smooth convex bodies

Intrinsic volumes of inscribed random polytopes in smooth convex bodies

Nash equilibria in random games

Random Polytopes, Convex Bodies, and Approximation

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