Christoffel and Minkowski problems in Minkowski space

Abstract

For convex sets in the Lorentzian Minkowski space bounded by space-like hyperplanes, it is possible to define area measures, similarly to the classical definition for convex bodies in the Euclidean space. Here the measures are defined on the hyperbolic space rather than on the round sphere. We are particularly interested by convex sets invariant under the action of isometries groups of the Minkowski space, so that the measures can be defined on compact hyperbolic manifolds. We can then look at the Christoffel and the Minkowski problems (i.e. particular measures are prescribed) in a general setting. In dimension (2+1), the Christoffel problem include a famous construction by G. Mess. In this dimension, the smooth version of the Minkowski problem already had a positive answer, and we show that this is a specificity of dimension (2+1), while the general problem has a solution in all dimensions.