Maximal Surface Area of Polytopes with Respect to Log-concave Rotation Invariant Measures

It was shown in [21] that the maximal surface area of a convex set in R with respect to a rotation invariant logconcave probability measure γ is of order √ n 4 √ V ar|X| √ E|X| , where X is a random vector in R 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) . √ n E|X| · √ logK · log n for all K. This bound is better then the general bound for all K ∈ [2, e c √ V ar|X| ]. Moreover, for all K in that range the bound is exact up to a factor of log n: for each K ∈ [2, e c √ V ar|X| ] there exists a polytope PK with at most K facets such that γ(∂PK) & √ n E|X| √ logK.