The Santaló point for the Holmes-Thompson boundary area

Abstract

We explore the notion of Santaló point for the Holmes-Thompson boundary area of a convex body in a normed space. In the case where the norm is $C^1$, and in the case where unit ball and convex body coincide, we prove existence and uniqueness. When the normed space has a smooth positively curved unit ball, we exhibit a dual Santaló point expressed as an average of centroids of projections of the dual body.

21 pages, 2 figures. We have added a new result to the paper showing that our previous uniqueness results hold without any regularity assumptions in case the two convex bodies B and K coincide. This is the content of the new Theorem 1.2