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Polytopes of Maximal Volume Product

Banff International Research Station for Mathematical Innovation and Discovery

For a convex body $K \subset \R^n$, let $$K^z = \{y\in \R^n : \langle y-z, x-z\rangle\le 1, \mbox{\ for all\ } x\in K\}$$ be the polar body of $K$ with respect to the center of polarity $z \in \R^n$. In this talk we would like to discuss the maximum of the volume product $$\mathcal{P}(K)=\min_{z\in {\rm int}(K)}|K||K^z|,$$ among convex polytopes $K\subset {\mathbb R}^n$ with a number of vertices bounded by some fixed integer $m \ge n+1$. In particular, we will show that the supremum is reached at a simplicial polytope with exactly $m$ vertices and we provide a new proof of a result of Meyer and Reisner showing that, in the plane, the regular polygon has maximal volume product among all polygons with at most $m$ vertices.

Mathematics; Functional analysis; Convex and discrete geometry

video/mp4

Author affiliation: Kent State University

2018-09-23

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/67232

Unreviewed

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