Hyperbolic geometry of shapes of convex bodies

Abstract

We use the intrinsic area to define a distance on the space of homothety classes of convex bodies in the n-dimensional Euclidean space, 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, and Alexandrov-Fenchel inequality is interpreted as the Lorentzian reversed Cauchy-Schwarz inequality. We deduce that the space of similarity classes of convex bodies has a proper geodesic distance with curvature bounded from below by 1 (in the sense of Alexandrov). In dimension 3, this space is homeomorphic to the space of distances with non-negative curvature on the 2-sphere, and this latter space contains the space of flat metrics on the 2-sphere considered by W. P. Thurston. Both Thurston's and the area distances rely on the area form. So the latter may be considered as a generalization of the "real part" of Thurston's construction.