Extremal general affine surface areas

Abstract

For a convex body K in Rn, we introduce and study the extremal general affine surface areas, defined byISφ(K):=supK′⊂K⁡asφ(K),osψ(K):=infK′⊃K⁡asψ(K) where asφ(K) and asψ(K) are the Lφ and Lψ affine surface area of K, respectively. We prove that there exist extremal convex bodies that achieve the supremum and infimum, and that the functionals ISφ and osψ are continuous. In our main results, we prove Blaschke-Santaló type inequalities and inverse Santaló type inequalities for the extremal general affine surface areas. This article may be regarded as an Orlicz extension of the recent work of Giladi, Huang, Schütt and Werner (2020), who introduced and studied the extremal Lp affine surface areas.