Random approximation and the vertex index of convex bodies

TL;DR

This paper demonstrates that a small random subset of a convex body can approximate the entire body with high probability, leading to a quadratic bound on the vertex index of convex bodies in high dimensions.

Contribution

It establishes a probabilistic approximation result for convex bodies and extends bounds on the vertex index to general convex bodies, not just symmetric ones.

Findings

Random subsets of size proportional to dimension approximate convex bodies.

High probability inclusion of the convex body within a scaled convex hull of the subset.

Vertex index of any convex body is bounded by a quadratic function of dimension.

Abstract

We prove that there exists an absolute constant $\alpha >1$ with the following property: if $K$ is a convex body in ${\mathbb R}^n$ whose center of mass is at the origin, then a random subset $X\subset K$ of cardinality ${\rm card}(X)=\lceil\alpha n\rceil $ satisfies with probability greater than $1-e^{-n}$ {K\subseteq c_1n\,{\mathrm conv}(X),} where $c_1>0$ is an absolute constant. As an application we show that the vertex index of any convex body $K$ in ${\mathbb R}^n$ is bounded by $c_2n^2$, where $c_2>0$ is an absolute constant, thus extending an estimate of Bezdek and Litvak for the symmetric case.