Speaker
Jean-Marc Schlenker
(Université du Luxembourg)
Description
Steiner asked in 1832 what are the combinatorial types of convex
polyhedra in $\R^3$ (or in $\R P^3$) 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.
