Constrained convex bodies with extremal affine surface areas

Abstract

Given a convex body K ⊆ R n and p ∈ R, we introduce and study the extremal inner and outer affine surface areaswhere 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 [22] 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. *

Extremal general affine surface areas

“…Our first result shows that for any convex body K in R n , the extremal affine surface areas exist if ϕ ∈ Conc * n (0, ∞) or ψ ∈ Conv * n (0, ∞). The analogous result was proved for the extremal p-affine surface areas in [5].…”

“…These definitions were motivated by those in [5], where the extremal L p affine surface areas were introduced (see Subsection 7.1 for more information). Since the L ϕ and L ψ affine surface areas have a "0/∞ property" on polytopes, we might expect the same for some of the extremal affine surface areas.…”

“…We shall henceforth only be interested in the functionals IS ϕ , OS ϕ * and os ψ on the classes Conc * n (0, ∞), Conc * n (0, ∞) and Conv * n (0, ∞), respectively. As remarked in [5] for the extremal p-affine surface areas, one may consider the inner maximal affine surface area IS ϕ (K) and the outer minimal affine surface area os ψ (K) as parallel notions to the John and Löwner ellipsoids of K, respectively.…”

“…Recently, Giladi, Huang, Schütt and Werner [5] defined the inner and outer maximal affine surface areas by…”

“…Recently, the extremal p-affine surface areas were introduced and studied by Giladi, Huang, Schütt and Werner [5]. Among these are the inner maximal p-affine surface area, defined to be the supremum of the p-affine surface areas of the convex bodies contained in K, and the outer minimal p-affine surface area, defined to by the infimum of the p-affine surface areas of the bodies that contain K. The purpose of this article is to extend those definitions to the setting of general affine surface areas, and to prove affine isoperimetric inequalities for these new extremal general affine surface areas.…”