Shapes of polyhedra, mixed volumes, and hyperbolic geometry
par
Ivan Izmestiev(Université de Fribourg)
→
Europe/Paris
Description
By a result of Bavard and Ghys, the space of Euclidean n-gons with fixed side directions can be viewed as an (n-3)-dimensional convex hyperbolic polyhedron. (This space has a natural linear structure; a vanishing side length corresponds to a hyperplane that bounds the polyhedron; the hyperbolic metric comes from a quadratic form that computes the area of the polygon.)
In this talk, based on a joint work with François Fillastre, I will present a generalization of the Bavard-Ghys construction to higher dimensions. We will study the space of Euclidean polyhedra with fixed facet normals. The combinatorial part of the construction deals with Gale diagrams, regular (aka coherent) triangulations, and with secondary polyhedra. The geometric part deals with mixed volumes and with Minkowski and Alexandrov-Fenchel inequalities.
