Higher Schläfli Formulas and Applications
Source: Compositio Mathematica, Volume 135, Number 1, January 2003 , pp. 1-24(24)
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.
Document Type: Research article
Affiliations: 1: Laboratoire Emile Picard, UMR CNRS 5580, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France. e-mail: email@example.com 2: Institut de Mathématiques de Jussieu, CNRS UMR 7586, Université Paris 7, Case 7012, 2 place Jussieu, 75251 Paris Cedex 05, France. e-mail: firstname.lastname@example.org
Publication date: 2003-01-01