On mean outer radii of random polytopes

In this paper we introduce a new sequence of quantities for random polytopes. Let $K_N=\text{conv}\{X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_N\}$ 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 also the right order of the expected value of ${\tilde{R}}_k(K_N)$ in the full range $n\leq N\leq e^{\sqrt{n}}$.