The Santal\'o point for the Holmes-Thompson boundary area

TL;DR

This paper investigates the Santaló point for the Holmes-Thompson boundary area in convex bodies within normed spaces, establishing existence, uniqueness, and explicit dual descriptions under certain smoothness conditions.

Contribution

It introduces the concept of the Santaló point for Holmes-Thompson boundary area and proves its existence and uniqueness in specific smooth normed spaces, providing explicit dual characterizations.

Findings

Existence and uniqueness of the Santaló point in smooth $C^1$ normed spaces.

Explicit dual Santaló point expressed as an average of centroids.

Results applicable when the unit ball and convex body coincide.

Abstract

We explore the notion of Santal\'o 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 Santal\'o point expressed as an average of centroids of projections of the dual body.