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 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.
| Original language | English |
|---|---|
| Pages (from-to) | 1221-1244 |
| Number of pages | 24 |
| Journal | Discrete and Computational Geometry |
| Volume | 70 |
| Issue number | 4 |
| DOIs | |
| State | Published - Dec 2023 |
Bibliographical note
Publisher Copyright:© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Funding
The authors thank Dominik Schmid for introducing them to the Pizza Theorem, and Ramon van Handel for dispelling some of their misconceptions and lending them a copy of []. They made extensive use of Geogebra to understand the 2-dimensional situation and to produce some of the figures. This work was partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR), and by Princeton University. This work was also partially supported by grants from the Simons Foundation (#429370 and #854548 to Richard Ehrenborg and #422467 to Margaret Readdy). The third author was also supported by NSF grant DMS-2247382. The authors thank Dominik Schmid for introducing them to the Pizza Theorem, and Ramon van Handel for dispelling some of their misconceptions and lending them a copy of [21]. They made extensive use of Geogebra to understand the 2-dimensional situation and to produce some of the figures. This work was partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR), and by Princeton University. This work was also partially supported by grants from the Simons Foundation (#429370 and #854548 to Richard Ehrenborg and #422467 to Margaret Readdy). The third author was also supported by NSF grant DMS-2247382.
| Funders | Funder number |
|---|---|
| LABEX MILYON | ANR-10-LABX-0070 |
| Ramon van Handel | |
| National Science Foundation (NSF) | DMS-2247382 |
| Simons Foundation | 854548, 422467, 429370 |
| Princeton University | |
| Agence Nationale de la Recherche | |
| Université de Lyon | ANR-11-IDEX-0007 |
Keywords
- 2-Structures
- Bolyai–Gerwien Theorem
- Coxeter arrangements
- Dissections
- Intrinsic volumes
- Pizza theorem
- Pseudo-root systems
- Reflection groups
ASJC Scopus subject areas
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics