  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.
