On mean outer radii of random polytopes

TL;DR

This paper introduces a new sequence of quantities called mean outer radii for random polytopes generated inside isotropic convex bodies, establishing their order and expected value behavior in high-dimensional probability.

Contribution

The paper defines the $k$-th mean outer radius for random polytopes and determines its order and expectation across a broad range of sample sizes.

Findings

Mean outer radii have order $ ext{max}\{ extsqrt{k}, extsqrt{ extlog N} angle L_K$ with high probability.

Expected mean outer radii match the high probability order in the full range of N.

Results hold for random polytopes with N between $n^2$ and $e^{ extsqrt{n}}$.

Abstract

In this paper we introduce a new sequence of quantities for random polytopes. Let $K_N=\conv\{X_1,...,X_N\}$ be a random polytope generated by independent random vectors uniformly distributed in an isotropic convex body $K$ of $\R^n$. We prove that the so-called $k$-th mean outer radius $\widetilde R_k(K_N)$ 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 also the right order of the expected value of $\widetilde R_k(K_N)$ in the full range $n\leq N\leq e^{\sqrt{n}}$.