On the isotropic constant of random polytopes with vertices on an $\ell_p$-sphere

TL;DR

This paper proves that random polytopes formed from points on the $ ext{l}_p$-sphere have bounded isotropic constants with high probability, extending previous results from the Euclidean case to general $p$-spheres.

Contribution

It generalizes the known Euclidean sphere results to $ ext{l}_p$-spheres, establishing uniform bounds on isotropic constants for a broader class of random polytopes.

Findings

Random polytopes have uniformly bounded isotropic constants with high probability.

The result extends Euclidean sphere findings to $ ext{l}_p$-spheres for $1 \\leq p < \infty$.

The proof combines probabilistic representations and moment estimates for sums of log-concave tail variables.

Abstract

The symmetric convex hull of random points that are independent and distributed according to the cone probability measure on the $\ell_p$-unit sphere of $\mathbb R^n$ for some $1\leq p < \infty$ is considered. We prove that these random polytopes have uniformly absolutely bounded isotropic constants with overwhelming probability. This generalizes the result for the Euclidean sphere ($p=2$) obtained by D. Alonso-Guti\'errez. The proof requires several different tools including a probabilistic representation of the cone measure due to G. Schechtman and J. Zinn and moment estimates for sums of independent random variables with log-concave tails originating in the work of E. Gluskin and S. Kwapie\'n.