Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T06:45:57.744Z Has data issue: false hasContentIssue false
  • English
  • Français

Weighted Brianchon-Gram Decomposition

Published online by Cambridge University Press:  20 November 2018

J. Agapito
J. Agapito*
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give in this note a weighted version of Brianchon and Gram's decomposition for a simple polytope. We can derive from this decomposition the weighted polar formula of Agapito and a weighted version of Brion's theorem, in a manner similar to Haase, where the unweighted case is worked out. This weighted version of Brianchon and Gram's decomposition is a direct consequence of the ordinary Brianchon–Gram formula.

Keywords

52B
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 2 , 01 June 2006 , pp. 161 - 169
Copyright
Copyright © Canadian Mathematical Society 2006

References

[A] Agapito, J., A weighted version of quantization commutes with reduction for a toric manifold. In: Integer Points in Polyhedra: Geometry, Number Theory, Algebra, Optimization, Contemporary Mathematics 374, American Mathematical Society, Providence, RI, 2005, pp. 1–14.Google Scholar
[B] Barvinok, A., A Course in Convexity. Graduate Studies in Mathematics 54, American Mathematical Society, Providence, RI, 2002.Google Scholar
[Br] Brianchon, C. J., Théorème nouveau sur les polyèdres. J. École Polytechnique, 15(1837), 317–319.Google Scholar
[Br1] Brion, M., Points entiers dans les polyèdres convexes. Ann. Sci. École Norm. Sup. (4) 21(1988), 653–663.Google Scholar
[Br2] Brion, M., Polyèdres et réseaux, Enseign. Math (2) 38(1992), no. 1-2, 71–88.Google Scholar
[G] Gram, J. P., Om rumvinklerne i et polyeder. Tidsskrift for Math. (Copenhagen) (3) 4(1874), 161–163.Google Scholar
[H] Haase, C., Polar decomposition and Brion's theorem. In: Integer Points in Polyhedra: Geometry, Number Theory, Algebra, Optimization, Contemporary Mathematics 374, American Mathematical Society, Providence, RI, 2005, pp. 91–99.Google Scholar
[L] Lawrence, J., Polytope volume computation. Math. Comp. 57(1991), no. 195, 259–271.Google Scholar
[S] Shephard, G. C., An elementary proof of Gram's theorem for convex polytopes. Canad. J. Math. 19(1967), 1214–1217.Google Scholar
[V] Varchenko, A. N., Combinatorics and topology of the arrangement of affine hyperplanes in the real space. (English translation) Funct. Anal. Appl. 21(1987), no. 1, 9–19.Google Scholar