Maximal surface area of polytopes with respect to log-concave rotation invariant measures

Abstract

It was shown in [19] that the maximal surface area of a convex set in Rn with respect to a rotation invariant log-concave probability measure γ is of order nVar|X|4E|X|, where X is a random vector in Rn distributed with respect to γ. In the present paper we discuss surface area of convex polytopes PK with K facets. We find tight bounds on the maximal surface area of PK in terms of K. We show that γ(∂PK)≲nE|X|⋅log⁡K⋅log⁡n for all K. This bound is better than the general bound for all K∈[2,ecVar|X|]. Moreover, for all K in that range the bound is exact up to a factor of log⁡n: for each K∈[2,ecVar|X|] there exists a polytope PK with at most K facets such that γ(∂PK)≳nE|X|log⁡K.