Pizza and 2-structures
Résumé
Let H be a Coxeter hyperplane arrangement on Rn with Coxeter group W. We assume that the map −id is in the group W and that H is not of type An1. Let K be a measurable subset of Rn with finite volume such that K is stable by W and let a∈Rn such that K contains the convex hull of {w(a):w∈W}. The authors proved in a previous article that the alternating sum over the chambers T of H of the volumes of the sets 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 with the volume replaced by any valuation that is invariant under affine isometries. This includes the cases of all intrinsic volumes. Apart from some basic geometry, the main ingredient is a previous theorem of the authors (the proof of which uses only properties of Coxeter systems and closed convex polyhedral cones) 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.
Domaines
Combinatoire [math.CO]
Origine : Fichiers produits par l'(les) auteur(s)