Hyperbolic geometry of shapes of convex bodies
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.