Constrained convex bodies with extremal affine surface areas
Abstract
Given a convex body K in Rn and p in R, we introduce and study the extremal inner and outer affine surface areas ISp(K) = supK'⊂eq K (asp(K') ) and osp(K)=infK'⊃eq K (asp(K') ), where asp(K') denotes the Lp-affine surface area of K', and the supremum is taken over all convex subsets of K and the infimum over all convex compact subsets containing K. The convex body that realizes IS1(K) in dimension 2 was determined by Barany. He also showed that this body is the limit shape of lattice polytopes in K. In higher dimensions no results are known about the extremal bodies. We use a thin shell estimate of Guedon and Milman and the L\"owner ellipsoid to give asymptotic estimates on the size of ISp(K) and osp(K). Surprisingly, both quantities are proportional to a power of volume.