## Sharing pizza in $n$ dimensions

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- by Richard Ehrenborg, Sophie Morel and Margaret Readdy PDF
- Trans. Amer. Math. Soc.
**375**(2022), 5829-5857 Request permission

## Abstract:

We introduce and prove the $n$-dimensional Pizza Theorem: Let $\mathcal {H}$ be a hyperplane arrangement in $\mathbb {R}^{n}$. 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 $\mathcal {H}$. We prove that if $\mathcal {H}$ is a Coxeter arrangement different from $A_{1}^{n}$ such that the group of isometries $W$ generated by the reflections in the hyperplanes of $\mathcal {H}$ contains the map $-\mathrm {id}$, 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 $\mathcal {H}$ 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 $0\in K+a$. 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 $R\geq \|a\|$ vanishes for a Coxeter arrangement $\mathcal {H}$ with $|\mathcal {H}|-n$ an even positive integer. We also prove the Pizza Theorem for the surface volume: When $\mathcal {H}$ is a Coxeter arrangement and $|\mathcal {H}| - n$ 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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## Additional Information

**Richard Ehrenborg**- Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
- MR Author ID: 312322
- ORCID: 0000-0001-5854-3890
- Email: richard.ehrenborg@uky.edu
**Sophie Morel**- Affiliation: Department of Mathematics and Statistics, ENS de Lyon, Unité De Mathématiques Pures Et Appliquées, 69342 Lyon Cedex 07, France
- MR Author ID: 824326
- ORCID: 0000-0001-9921-7696
- Email: sophie.morel@ens-lyon.fr
**Margaret Readdy**- Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
- MR Author ID: 363918
- ORCID: 0000-0002-3648-0865
- Email: margaret.readdy@uky.edu
- Received by editor(s): February 12, 2021
- Received by editor(s) in revised form: February 8, 2022
- Published electronically: June 3, 2022
- Additional Notes: This work was partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR), and by Princeton University. The third author also thanks Princeton University for hosting a one-week visit in Spring 2020, and the second author thanks the University of Kentucky for its hospitality during a one-week visit in Fall 2019. This work was also partially supported by grants from the Simons Foundation (#429370 to the first author and #422467 to the third author).
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 5829-5857 - MSC (2020): Primary 51F15, 52C35, 51M20, 51M25; Secondary 26B15
- DOI: https://doi.org/10.1090/tran/8664
- MathSciNet review: 4469238