Géométrie et Topologie

Polyhedra inscribed in quadrics

Non programmé

50m

Orateur

Jean-Marc Schlenker (Université du Luxembourg)

Description

Steiner asked in 1832 what are the combinatorial types of convex polyhedra in $\text{Undefined control sequence R}$ (or in $\text{Undefined control sequence R}$) with all their vertices on a quadric. An answer was given in 1990 by Hodgson, Rivin and Smith for polyhedra inscribed in a sphere, that is, contained in a ball and with all their vertices on its boundary. We will describe a similar result (obtained with Jeff Danciger and Sara Maloni) for polyhedra inscribed in a one-sheeted hyperboloid or a cylinder. Steiner's question also asks about polyhedra {\em weakly} inscribed in a quadric, that is, with vertices on the quadric but not entirely on one side. We will also describe the possible combinatorics of polyhedra weakly inscribed in a sphere (joint with Hao Chen). The proofs are based on hyperbolic, de Sitter and anti-de Sitter geometry.