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Volume product and metric spaces

Banff International Research Station for Mathematical Innovation and Discovery

Given a finite metric space M, the set of Lipschitz functions on M with Lipschitz constant at most 1 can be identified with a convex body, say K(M), in R^n. The volume product P(M)=|K(M)| |K(M)º| is an isometric invariant of M. We discuss the extreme properties of the volume product. We show that if P(M) is maximal among all the metric spaces with the same number of points, then all triangle inequalities in M are strict and K(M)º is simplicial. We also focus on the metric spaces minimizing the volume product, and in the Mahler's conjecture for this class of convex bodies. In addition, we characterize the metric spaces such that K(M) is a Hanner polytope. This is a joint work with M. Alexander, M. Fradelizi, and A. Zvavitch.

Mathematics; Convex and discrete geometry; Functional analysis

video/mp4

Author affiliation: Kent State University

2020-08-12

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/75425

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/