Hyperbolic geometry of shapes of convex bodies

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