## Abstract

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.

Original language | English |
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Pages (from-to) | 5829-5857 |

Number of pages | 29 |

Journal | Transactions of the American Mathematical Society |

Volume | 375 |

Issue number | 8 |

DOIs | |

State | Published - Aug 1 2022 |

### Bibliographical note

Funding Information: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). The authors thank Dominik Schmid for introducing us to the Pizza Theorem. We also thank Theodore Ehrenborg and the referee for their comments on the manuscript.

Funding Information:

This work was partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Univer-sité 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).

Publisher Copyright:

© 2022 American Mathematical Society.

## ASJC Scopus subject areas

- Mathematics (all)
- Applied Mathematics