Seminars & Colloquia Calendar

Title: Sharing Pizza in n Dimensions Speaker: Richard Ehrenborg, University of Kentucky Date: Thursday, March 31st, 2022 Time: 5:00pm–5:48pm Place: zoom Short abstract: We introduce and prove the n-dimensional Pizza Theorem. This is joint work with Sophie Morel and Margaret Readdy. Long abstract: We introduce and prove the n-dimensional Pizza Theorem. Let H be a real ndimensional 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 An 1 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.