Pizza and 2-Structures
Tóm tắt
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.
Tài liệu tham khảo
citation_title=Combinatorics of Coxeter Groups; citation_publication_date=2005; citation_id=CR1; citation_author=A Björner; citation_author=F Brenti; citation_publisher=Springer
Boltianskiĭ, V. G.: Hilbert’s third problem. V. H. Winston & Sons, Washington, DC.; Halsted Press [Wiley], New York-Toronto-London (1978). Translated from the Russian by Richard A. Silverman, With a foreword by Albert B. J. Novikoff, Scripta Series in Mathematics
Bourbaki, N.: Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines. Actualités Scientifiques et Industrielles, No. 1337. Hermann, Paris (1968)
citation_journal_title=St. Petersbg. Math. J.; citation_title=Reflection Groups and the Pizza Theorem; citation_author=YuA Brailov; citation_volume=33; citation_issue=6; citation_publication_date=2022; citation_pages=891-896; citation_doi=10.1090/spmj/1732; citation_id=CR4
citation_journal_title=Math. Mag.; citation_title=Proof without words: fair allocation of a pizza; citation_author=L Carter, S Wagon; citation_volume=67; citation_issue=4; citation_publication_date=1994; citation_pages=267; citation_doi=10.1080/0025570X.1994.11996228; citation_id=CR5
citation_journal_title=J. Comb. Algebra; citation_title=A generalization of combinatorial identities for stable discrete series constants; citation_author=R Ehrenborg, S Morel, M Readdy; citation_volume=6; citation_issue=1; citation_publication_date=2022; citation_pages=109-183; citation_doi=10.4171/JCA/62; citation_id=CR6
citation_journal_title=Trans. Am. Math. Soc.; citation_title=Sharing pizza in $n$ dimensions; citation_author=R Ehrenborg, S Morel, M Readdy; citation_volume=375; citation_issue=8; citation_publication_date=2022; citation_pages=5829-5857; citation_id=CR7
Euclid. Elements. Book 1, c. 300 BC
citation_journal_title=Math. Mag.; citation_title=The proof is in the pizza; citation_author=GN Frederickson; citation_volume=85; citation_issue=1; citation_publication_date=2012; citation_pages=26-33; citation_doi=10.4169/math.mag.85.1.26; citation_id=CR9
citation_journal_title=Math. Mag.; citation_title=Divisors of a circle: solution to problem 660; citation_author=M Goldberg; citation_volume=41; citation_issue=1; citation_publication_date=1968; citation_pages=46; citation_id=CR10
citation_journal_title=Pac. J. Math.; citation_title=On the extension of additive functionals on classes of convex sets; citation_author=H Groemer; citation_volume=75; citation_issue=2; citation_publication_date=1978; citation_pages=397-410; citation_doi=10.2140/pjm.1978.75.397; citation_id=CR11
citation_title=Finite reflection groups; citation_publication_date=1985; citation_id=CR12; citation_author=LC Grove; citation_author=CT Benson; citation_publisher=Springer
Herb, R.A.: Two-structures and discrete series character formulas. In: The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998), volume 68 of Proceedings of Symposia in Pure Mathematics pp. 285–319. American Mathematical Society, Providence, RI (2000)
citation_journal_title=Trans. Am. Math. Soc.; citation_title=Discrete series characters as lifts from two-structure groups; citation_author=RA Herb; citation_volume=353; citation_issue=7; citation_publication_date=2001; citation_pages=2557-2599; citation_doi=10.1090/S0002-9947-01-02827-6; citation_id=CR14
citation_journal_title=Austral. Math. Soc. Gaz.; citation_title=The pizza theorem; citation_author=J Hirschhorn, M Hirschhorn, JK Hirschhorn, A Hirschhorn, P Hirschhorn; citation_volume=26; citation_issue=3; citation_publication_date=1999; citation_pages=120-121; citation_id=CR15
citation_title=Reflection Groups and Coxeter Groups; citation_publication_date=1990; citation_id=CR16; citation_author=JE Humphreys; citation_publisher=Cambridge University Press
citation_journal_title=Math. Scand.; citation_title=The algebra of polytopes in affine spaces; citation_author=B Jessen, A Thorup; citation_volume=43; citation_publication_date=1978; citation_pages=211-240; citation_doi=10.7146/math.scand.a-11777; citation_id=CR17
citation_title=Introduction to Geometric Probability. Lezioni Lincee. [Lincei Lectures].; citation_publication_date=1997; citation_id=CR18; citation_author=DA Klain; citation_author=GC Rota; citation_publisher=Cambridge University Press
citation_journal_title=Am. Math. Monthly; citation_title=Of cheese and crust: a proof of the pizza conjecture and other tasty results; citation_author=R Mabry, P Deiermann; citation_volume=116; citation_issue=5; citation_publication_date=2009; citation_pages=423-438; citation_doi=10.1080/00029890.2009.11920956; citation_id=CR19
citation_title=Hilbert’s Third Problem: Scissors Congruence; citation_publication_date=1979; citation_id=CR20; citation_author=CH Sah; citation_publisher=Pitman. (Advanced Publishing Program)
citation_title=Convex Bodies: The Brunn–Minkowski Theory; citation_publication_date=2014; citation_id=CR21; citation_author=R Schneider; citation_publisher=Cambridge University Press
citation_title=Stochastic and Integral Geometry; citation_publication_date=2008; citation_id=CR22; citation_author=R Schneider; citation_author=W Weil; citation_publisher=Springer