The isotropic constant of random polytopes with vertices on convex surfaces

TL;DR

This paper establishes bounds on the isotropic constant of random polytopes formed from points on the boundary of convex bodies, showing it is typically small, especially for unconditional bodies, using concentration inequalities and new estimates.

Contribution

It provides new probabilistic bounds on the isotropic constant of random polytopes generated by boundary points, including a novel $ ext{psi}_2$-estimate for cone measures.

Findings

With high probability, $L_{K_N} \

For unconditional bodies, $L_{K_N} \

New $ ext{psi}_2$-estimate for cone measure functionals.

Abstract

For an isotropic convex body $K\subset\mathbb{R}^n$ we consider the isotropic constant $L_{K_N}$ of the symmetric random polytope $K_N$ generated by $N$ independent random points which are distributed according to the cone probability measure on the boundary of $K$. We show that with overwhelming probability $L_{K_N}\leq C\sqrt{\log(2N/n)}$, where $C\in(0,\infty)$ is an absolute constant. If $K$ is unconditional we argue that even $L_{K_N}\leq C$ with overwhelming probability. The proofs are based on concentration inequalities for sums of sub-exponential or sub-Gaussian random variables, respectively, and, in the unconditional case, on a new $\psi_2$-estimate for linear functionals with respect to the cone measure in the spirit of Bobkov and Nazarov, which might be of independent interest.