Fuchsian convex bodies: basics of Brunn--Minkowski theory

Details Fuchsian convex bodies: basics of Brunn--Minkowski theory Fuchsian convex bodies: basics of Brunn--Minkowski theory

2011-12-22

arxiv.org/abs/1112.5353v1

Mathematics

Differential Geometry

Scientific paper

The hyperbolic space \mathbb{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 \mathbb{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 defined, 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 volume (it is log concave in the classical setting). This implies analogs of Alexandrov--Fenchel and Brunn--Minkowski inequalities. Here the inequalities are reversed.

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