This is an announcement for the paper "On mean outer radii of random polytopes" by David Alonso-Gutierrez, Nikos Dafnis, Maria A. Hernandez Cifre, and Joscha Prochno.

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 $\R^n$. 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 $n\leq N\leq e^{\sqrt{n}}$.

Archive classification: math.FA

Mathematics Subject Classification: Primary 52A22, Secondary 52A23, 05D40

Remarks: 14 pages

Submitted from: prochno@math.uni-kiel.de

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1211.2336

or

http://arXiv.org/abs/1211.2336