The Gauss curvature of a convex body can be seen as a measure on the unit sphere (with some properties). For such a measure μ , the Alexandrov problem consists in proving the existence and uniqueness of a convex body whose curvature measure is μ.

In the Euclidean space, this problem was first solved by Alexandrov, and it was observed later that it is equivalent to an optimal transport problem on the sphere. In this talk I will consider Alexandrov problem for convex bodies of the hyperbolic space. After defining the curvature measure, I will explain what are its main properties. If time permits, I will explain how the optimal transport approach of Alexandrov problem leads to a non-linear Kantorovich problem on the sphere, and how to solve it.

Joint work with Jérôme Bertrand.