Inner products for Convex Bodies
Metric Geometry
2018-12-14 v2
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.
Keywords
Cite
@article{arxiv.1811.03686,
title = {Inner products for Convex Bodies},
author = {David Bryant and Petru Cioica-Licht and Lisa Orloff Clark and Rachael Young},
journal= {arXiv preprint arXiv:1811.03686},
year = {2018}
}