SHARING PIZZA IN n DIMENSIONS

We introduce and prove the n-dimensional Pizza Theorem: Let H be a hyperplane arrangement in Rⁿ. 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 Aⁿ₁ 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.