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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 oflinear isometries of the Minkowski space such that $\H^d/\Gamma$ is a compactmanifold. We introduce Fuchsian convex bodies, which are closed convex sets inMinkowski space, globally invariant for the action of a Fuchsian group. Avolume can be associated to each Fuchsian convex body, and, if the group isfixed, Minkowski addition behaves well. Then Fuchsian convex bodies can bestudied in the same manner as convex bodies of Euclidean space in the classicalBrunn--Minkowski theory. For example, support functions can be defined, asfunctions on a compact hyperbolic manifold instead of the sphere.  The main result is the convexity of the associated volume (it is log concavein the classical setting). This implies analogs of Alexandrov--Fenchel andBrunn--Minkowski inequalities. Here the inequalities are reversed.

Fuchsian convex bodies: basics of Brunn--Minkowski theory