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Higher Schläfli Formulas and Applications

Published online by Cambridge University Press:  04 December 2007

Jean-Marc Schlenker  and
Rabah Souam
Jean-Marc Schlenker
Affiliation:
Laboratoire Emile Picard, UMR CNRS 5580, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France. e-mail: schlenker@picard.ups-tlse.fr
Rabah Souam
Affiliation:
Institut de Mathématiques de Jussieu, CNRS UMR 7586, Université Paris 7, Case 7012, 2 place Jussieu, 75251 Paris Cedex 05, France. e-mail: souam@math.jussieu.fr
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Abstract

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The classical Schläfli formula relates the variations of the dihedral angles of a smooth family of polyhedra in a space-form to the variation of the enclosed volume. We give higher analogues of this formula: for each p, we prove a simple formula relating the variation of the volumes of the codimension p faces to the variation of the ‘curvature’ – the volumes of the duals of the links in the convex case – of codimension p+2 faces. It is valid also for ideal polyhedra, or for polyhedra with some ideal vertices. This extends results of Suárez-Peiró. The proof is through analoguous smooth formulas. Some applications are described.

Keywords

mean curvaturepolyhedraSchläflispace-forms
Type
Research Article
Information
Compositio Mathematica , Volume 135 , Issue 1 , January 2003 , pp. 1 - 24
Copyright
© 2003 Kluwer Academic Publishers