Maximal surface area of a convex set in with respect to exponential rotation invariant measures

Let p be a positive number. Consider the probability measure γ p with density φ p ( y ) = c n , p e − | y | p p . We show that the maximal surface area of a convex body in R n with respect to γ p is asymptotically equivalent to C ( p ) n 3 4 − 1 p , where the constant C ( p ) depends on p only. This is a generalization of results due to Ball (1993) [1] and Nazarov (2003) [9] in the case of the standard Gaussian measure γ 2 .