Random approximation and the vertex index of convex bodies

Abstract

We prove that there exists an absolute constant α > 1 with the following property: if K is a convex body in R n whose center of mass is at the origin, then a random subset X ⊂ K of cardinality card(X) = ⌈αn⌉ satisfies with probability greater than 1 − e −n K ⊆ c1n conv(X), where c1 > 0 is an absolute constant. As an application we show that the vertex index of any convex body K in R n is bounded by c2n 2 , where c2 > 0 is an absolute constant, thus extending an estimate of Bezdek and Litvak for the symmetric case.

Approximating a Convex Body by a Polytope Using the Epsilon-Net Theorem

“…The main result of [BCH16] concerns the case of very rough approximation, that is, where the number t of chosen points is linear in the dimension d. It states that the convex hull of t = αd random points in a centered convex body K is a polytope P which satisfies c 1 d K ⊆ P , with probability 1 − δ = 1 − e −c 2 d , where c 1 , c 2 > 0 and α > 1 are absolute constants.…”