Inner products for Convex Bodies

TL;DR

This paper introduces a new set inner product for convex bodies, extending classical geometric results and enabling a geometric framework for convex body analysis with applications in evolutionary biology.

Contribution

It defines a novel set inner product satisfying key properties and demonstrates its embedding into an inner product space, extending prior theoretical results.

Findings

Set inner product can be embedded into an inner product space on support functions

Provides a new geometric structure on convex bodies

Application to reconstruct ancestral ecological niches

Abstract

We define a set inner product to be a function on pairs of convex bodies which is symmetric, Minkowski linear in each dimension, positive definite, and satisfies the natural analogue of the Cauchy-Schwartz inequality (which is not implied by the other conditions). We show that any set inner product can be embedded into an inner product space on the associated support functions, thereby extending fundamental results of Hormander and Radstrom. The set inner product provides a geometry on the space of convex bodies. We explore some of the properties of that geometry, and discuss an application of these ideas to the reconstruction of ancestral ecological niches in evolutionary biology.