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Article Dans Une Revue Transactions of the American Mathematical Society Année : 2022

Sharing pizza in n dimensions

Résumé

We introduce and prove the n-dimensional Pizza Theorem: Let H be a hyperplane arrangement in R n. If K is a measurable set of finite volume, the pizza quantity of K is the alternating sum of the volumes of the regions obtained by intersecting K with the arrangement H. We prove that if H is a Coxeter arrangement different from A n 1 such that the group of isometries W generated by the reflections in the hyperplanes of H contains the map − id, and if K is a translate of a convex body that is stable under W and contains the origin, then the pizza quantity of K is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of H that we call the even restricted arrangement. More generally, we prove that for a class of arrangements that we call even (this includes the Coxeter arrangements above) and for a sufficiently symmetric set K, the pizza quantity of K+a is polynomial in a for a small enough, for example if K is convex and 0 ∈ K + a. We get stronger results in the case of balls, more generally, convex bodies bounded by quadratic hypersurfaces. For example, we prove that the pizza quantity of the ball centered at a having radius R ≥ a vanishes for a Coxeter arrangement H with |H| − n an even positive integer. We also prove the Pizza Theorem for the surface volume: When H is a Coxeter arrangement and |H| − n is a nonnegative even integer, for an n-dimensional ball the alternating sum of the (n − 1)-dimensional surface volumes of the regions is equal to zero.
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Dates et versions

hal-03450174 , version 1 (25-11-2021)
hal-03450174 , version 2 (23-11-2022)
hal-03450174 , version 3 (22-10-2023)

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  • HAL Id : hal-03450174 , version 2

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Richard Ehrenborg, Sophie Morel, Margaret Readdy. Sharing pizza in n dimensions. Transactions of the American Mathematical Society, 2022. ⟨hal-03450174v2⟩
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