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Polytopes of Maximal Volume Product

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Abstract

For a convex body \(K \subset {\mathbb {R}}^n,\) let \(K^z = \{y\in {\mathbb R}^n : \langle y-z, x-z\rangle \le 1,\ \text{ for } \text{ all }\ x\in K\}\) be the polar body of K with respect to the center of polarity \(z \in {\mathbb {R}}^n.\) The goal of this paper is to study the maximum of the volume product \(\mathcal {P}(K)=\min _{z\in \mathrm{int}(K)}|K||K^z|,\) 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 prove a combinatorial formula characterizing a polytope of maximal volume product and use this formula to show that the supremum is reached at a simplicial polytope with exactly m vertices. We also use this formula to provide a 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. Finally, we treat the case of polytopes with \(n+2\) vertices in \({\mathbb {R}}^n.\)

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Authors and Affiliations

  1. Department of Mathematics, Kent State University, Kent, OH, 44242, USA

    Matthew Alexander & Artem Zvavitch

  2. Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, 77454, Marne-la-Vallée, France

    Matthieu Fradelizi

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  1. Matthew Alexander
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Correspondence to Artem Zvavitch.

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A. Zvavitch is supported in part by the U.S. National Science Foundation Grant DMS-1101636; the Bézout Labex of Université Paris-Est and la Comue Université Paris-Est. M. Alexander is supported in part by the Chateaubriand Fellowship of the Office for Science and Technology of the Embassy of France in the United States and The Centre National de la Recherche Scientifique Funding Visiting Research at Université Paris-Est Marne-la-Vallée. M. Fradelizi is supported in part by the Agence Nationale de la Recherche, Project GeMeCoD (ANR 2011 BS01 007 01).

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Alexander, M., Fradelizi, M. & Zvavitch, A. Polytopes of Maximal Volume Product. Discrete Comput Geom 62, 583–600 (2019). https://doi.org/10.1007/s00454-019-00072-3

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  • DOI: https://doi.org/10.1007/s00454-019-00072-3

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