Constrained convex bodies with extremal affine surface areas

Given a convex body K⊆Rn and p∈R, we introduce and study the extremal inner and outer affine surface areasISp(K)=supK′⊆K⁡(asp(K′)) and osp(K)=infK′⊇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 in [3] where it was also shown 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 [23] and the Löwner ellipsoid to give asymptotic estimates on the size of ISp(K) and osp(K). Surprisingly, it turns out that both quantities are proportional to a power of volume.