20-24 November 2017
Europe/Paris timezone

Polyhedra inscribed in quadrics

Not scheduled


Jean-Marc Schlenker (Université du Luxembourg)


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.

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