On mean outer radii of random polytopes

Abstract

In this paper we introduce a new sequence of quantities for random polytopes. Let KN = conv{X1, . . . ,XN} be a random polytope generated by independent random vectors uniformly distributed in an isotropic convex body K of Rn. We prove that the so-called k-th mean outer radius e Rk(KN) has order max{ pk,plogN}LK with high probability if n2 N epn. We also show that this is also the right order of the expected value of e Rk(KN) in the full range n N epn.