Asymptotic shape of the convex hull of isotropic log-concave random vectors

TL;DR

This paper analyzes the asymptotic geometric properties of convex hulls formed by isotropic log-concave random vectors, providing sharp estimates for their geometric parameters across a broad range of sample sizes.

Contribution

It extends previous work by offering sharp estimates for geometric parameters of convex hulls for larger sample sizes, utilizing recent advances in centroid body analysis.

Findings

Sharp estimates for geometric parameters of convex hulls for N up to exp(n)

Extension of previous results to larger sample size ranges

Application of Milman's recent results on centroid bodies

Abstract

Let $x_1,\ldots ,x_N$ be independent random points distributed according to an isotropic log-concave measure $\mu $ on ${\mathbb R}^n$, and consider the random polytope $$K_N:={\rm conv}\{ \pm x_1,\ldots ,\pm x_N\}.$$ We provide sharp estimates for the querma\ss{}integrals and other geometric parameters of $K_N$ in the range $cn\ls N\ls\exp (n)$; these complement previous results from \cite{DGT1} and \cite{DGT} that were given for the range $cn\ls N\ls\exp (\sqrt{n})$. One of the basic new ingredients in our work is a recent result of E.~Milman that determines the mean width of the centroid body $Z_q(\mu )$ of $\mu $ for all $1\ls q\ls n$.