IUMJ

Title: On mean outer radii of random polytopes

Authors: Joscha Prochno, David Alonso-Gutierrez, Nikos Dafnis and Maria A. Hernandez-Cifre

Issue: Volume 63 (2014), Issue 2, 579-595

Abstract:

In this paper, we introduce a new sequence of quantities for random polytopes. Let $K_N=\conv{X_1,\dots,X_N\}$ be a random polytope generated by independent random vectors uniformly distributed in an isotropic convex body $K$ of $\mathbb{R}^n$. We prove that the so-called $k$-th mean outer radius $\tilde{R}_k(K_N)$ has order $\max\{\sqrt{k},\sqrt{\log N}\}L_K$ with high probability if $n^2\leq N\leq e^{\sqrt{n}}$. We also show that this is the right order of the expected value of $\tilde{R}_k(K_N)$ in the full range $n\leq N\leq e^{\sqrt{n}}$.