Sharing pizza in $n$ dimensions

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.