Hyperbolic geometry of shapes of convex bodies

Abstract

We define a distance on the space of convex bodies in the n-dimensional Euclidean space, up to translations and homotheties, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient Lorentzian structure is an extension of the intrinsic area form of convex bodies.We deduce that the space of shapes of convex bodies (i.e. convex bodies up to similarities) has a proper distance with curvature bounded from below by −1. In dimension 3, this space naturally identifies with the space of distances with non-negative curvature on the 2-sphere.