Über Tangentialkörper der Kugel

Abstract

The convex body K is called a p-tangential body of the convex body$\bar K$ if$\bar K \subseteq K$ and every (n−p−1)-extremal support plane of K is also a support plane of ¯K. It has been conjectured by Minkowski and proved by Bol that 1-tangential bodies of balls are characterized by the case of equality in one of the Minkowski inequalities for the quermassintegrals. This fact, together with a geometric description of the support of the surface area function S_p(K,·) of order p which was obtained earlier by the author, is used to prove some new characterizations of p-tangential bodies. For instance, an n-dimensional convex body K is a p-tangential body of a ball (for some p∈{0,..., n−2}) if, and only if, its surface area functions S_p (K,·) and S_n−1 (K,·) are constant multiples of each other.