Small-Ball Probabilities for the Volume of Random Convex Sets

Abstract

We prove small-deviation estimates for the volume of random convex sets. The focus is on convex hulls and Minkowski sums of line segments generated by independent random points. The random models considered include (Lebesgue) absolutely continuous probability measures with bounded densities and the class of log-concave measures.

Affine quermassintegrals of random polytopes

“…The left hand side of (1.6) was proved by Paouris and Pivovarov in [32]; it confirms (1.1) in an isomorphic sense. Theorem 1.1 (Paouris-Pivovarov).…”

Mean Width of Regular Polytopes and Expected Maxima of Correlated Gaussian Variables

“…It is possible to obtain further asymptotic terms in (17) and (18), (see, e.g., [15, Eq. (2.4.8) on p. 39]) but it seems that none of these expansions can correctly capture the very small difference between the expectations in Theorems 2.1 and 2.3.…”

Variants of the Busemann–Petty Problem and of the Shephard Problem

“…We also prove a general estimate, which is logarithmic in n and valid for all k. The proof is based on estimates from [25]. Theorem 1.8.…”

“…c 1 n/m |K| The left hand side of (4.17) was proved by Paouris and Pivovarov in[25]: Theorem 4.6 (Paouris-Pivovarov). Let A be a convex body in R n .…”