Maximal Surface Area of a convex set in $\R^n$ with respect to log concave rotation invariant measures

Abstract

It was shown by K. Ball and F. Nazarov, that the maximal surface area of a convex set in $\mathbb{R}^n$ with respect to the Standard Gaussian measure is of order $n^{\frac{1}{4}}$. In the present paper we establish the analogous result for all rotation invariant log concave probability measures. We show that the maximal surface area with respect to such measures is of order $\frac{\sqrt{n}}{\sqrt[4]{Var|X|} \sqrt{\mathbb{E}|X|}}$, where $X$ is a random vector in $\mathbb{R}^n$ distributed with respect to the measure.