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

Let $p$ be a positive number. Consider probability measure $γ_p$ with density $φ_p(y)=c_{n,p}e^{-\frac{|y|^p}{p}}$. We show that the maximal surface area of a convex body in $\mathbb{R}^n$ with respect to $γ_p$ 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 $γ_2$.

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