  Let $\mathcal{H}$ be a Coxeter hyperplane arrangement in $n$-dimensional
Euclidean space. Assume that the negative of the identity map belongs to the
associated Coxeter group $W$. Furthermore assume that the arrangement is not of
type $A_1^n$. Let $K$ be a measurable subset of the Euclidean space with finite
volume which is stable by the Coxeter group $W$ and let $a$ be a point such
that $K$ contains the convex hull of the orbit of the point $a$ under the group
$W$. In a previous article the authors proved the generalized pizza theorem:
that the alternating sum over the chambers $T$ of $\mathcal{H}$ of the volumes
of the intersections $T\cap(K+a)$ is zero. In this paper we give a dissection
proof of this result. In fact, we lift the identity to an abstract dissection
group to obtain a similar identity that replaces the volume by any valuation
that is invariant under affine isometries. This includes the cases of all
intrinsic volumes. Apart from basic geometry, the main ingredient is a theorem
of the authors where we relate the alternating sum of the values of certain
valuations over the chambers of a Coxeter arrangement to similar alternating
sums for simpler subarrangements called $2$-structures introduced by Herb to
study discrete series characters of real reduced groups.
