Discrete CATS Seminar

Speaker:  Richard Ehrenborg, University of Kentucky

Title:  Sharing Pizza in n Dimensions

Monday, February 7th 2022

2 pm POT 745

Abstract:

We introduce and prove the n-dimensional Pizza Theorem. Let ℋ be a real n-dimensional hyperplane arrangement. If K is a convex 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_1^n such that the group of isometries W generated by the reflections in the hyperplanes of ℋ contains the negative of the identity map, and if K is a translate of a convex set 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. We get stronger results in the case of balls. We prove that the pizza quantity of a ball containing the origin vanishes for a Coxeter arrangement ℋ with |ℋ|-n an even positive integer.



This is joint work with Sophie Morel and Margaret Readdy.