Hyperbolic geometry of shapes of convex bodies

TL;DR

This paper introduces a hyperbolic geometric framework for the space of convex bodies using intrinsic area, revealing new curvature properties and connections to Thurston's flat metrics on the sphere.

Contribution

It defines a hyperbolic distance on convex bodies via intrinsic area, extending the Lorentzian structure and linking to Thurston's metrics, with implications for curvature bounds.

Findings

The space of convex bodies is isometric to a convex subset of hyperbolic space.

The Lorentzian structure extends the intrinsic area form and interprets the Alexandrov--Fenchel Inequality.

The space has a proper geodesic distance with curvature bounded below by -1.

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.