Thin Shell Estimates for Maximal Affine Surface Area

Given a convex body K in R^n, we study the maximal affine surface area of K, i.e., the quantity AS(K) = sup_{C subseteq K} as(C) where as(C) denotes the affine surface area of C, and the supremum is taken over all convex subsets of K. In particular, we give asymptotic estimates on its size.