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- W2964031723 abstract "For a convex body $$K subset {mathbb {R}}^n,$$ let $$K^z = {yin {mathbb R}^n : langle y-z, x-zrangle le 1, text{ for } text{ all } xin K}$$ 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 $$mathcal {P}(K)=min _{zin mathrm{int}(K)}|K||K^z|,$$ among convex polytopes $$Ksubset {mathbb R}^n$$ 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 $${mathbb {R}}^n.$$" @default.
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- W2964031723 date "2019-03-18" @default.
- W2964031723 modified "2023-09-30" @default.
- W2964031723 title "Polytopes of Maximal Volume Product" @default.
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- W2964031723 doi "https://doi.org/10.1007/s00454-019-00072-3" @default.
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