Fuchsian convex bodies: basics of Brunn--Minkowski theory

TL;DR

This paper extends the classical Brunn--Minkowski theory to Fuchsian convex bodies in hyperbolic space, establishing volume convexity and deriving reversed inequalities analogous to Euclidean results.

Contribution

It introduces Fuchsian convex bodies in Minkowski space, develops their volume theory, and proves convexity and reversed inequalities similar to classical convex geometry.

Findings

Volume of Fuchsian convex bodies is convex (log concave)

Reversed Alexandrov--Fenchel inequalities hold

Reversed Brunn--Minkowski inequalities are established

Abstract

The hyperbolic space $ \H^d$ can be defined as a pseudo-sphere in the $(d+1)$ Minkowski space-time. In this paper, a Fuchsian group $\Gamma$ is a group of linear isometries of the Minkowski space such that $\H^d/\Gamma$ is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, globally invariant for the action of a Fuchsian group. A volume can be associated to each Fuchsian convex body, and, if the group is fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be studied in the same manner as convex bodies of Euclidean space in the classical Brunn--Minkowski theory. For example, support functions can be defined, as functions on a compact hyperbolic manifold instead of the sphere. The main result is the convexity of the associated volume (it is log concave in the classical setting). This implies analogs of Alexandrov--Fenchel and Brunn--Minkowski inequalities. Here the inequalities are reversed.