Journal of Convex Analysis 28 (2021), No. 4, 1249--1264
Copyright Heldermann Verlag 2021
Inner Products for Convex Bodies
David Bryant
Dept. of Mathematics and Statistics, University of Otago, Dunedin, New Zealand
david.bryant@otago.ac.nz
Petru Cioica-Licht
Fakultät für Mathematik, Universität Duisburg-Essen, Essen, Germany
petru.cioica-licht@uni-due.de
Lisa Orloff Clark
School of Mathematics and Statistics, Victoria University, Wellington, New Zealand
lisa.clark@vuw.ac.nz
Rachael Young
Dept. of Mathematics and Statistics, University of Otago, Dunedin, New Zealand
rachael.gray.young@gmail.com
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 Hörmander and Radström. 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: Inner product, convex body, Minkowski linear functionals, ecological niche.
MSC: 52A20, 52A27, 05C05.
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