Constrained convex bodies with maximal affine surface area

Abstract:

Given a convex body $K$ in $R^n$, we study the maximal affine surface area of $K$, i.e., the quantity $AS(K) = sup_{C} as(C)$ 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 the size of $AS(K)$.