Séminaire Lotharingien de Combinatoire, 89B.48 (2023), 12 pp.
Richard Ehrenborg, Sophie Morel and Margaret Readdy
Pizza and 2-Structures
Abstract.
Let 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 A1n.
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 H of
the volumes of the intersections T ∩ (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 previous 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.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
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