Maximal surface area of a convex set in ${\mathbb{R}}^n$ with respect to exponential rotation invariant measures

Abstract

Let $p$ be a positive number. Consider probability measure ${\gamma }_p$ with density ${\phi }_p(y)=c_{n,p}{e}^{-\frac{|y{|}^p}{p}}$. We show that the maximal surface area of a convex body in ${\mathbb{R}}^n$ with respect to ${\gamma }_p$ is asymptotically equal to $C_p{n}^{\frac{3}{4}-\frac{1}{p}}$, where constant $C_p$ depends on $p$ only. This is a generalization of Ball's and Nazarov's bounds, which were given for the case of the standard Gaussian measure ${\gamma }_2$.