  The Gauss curvature measure of a pointed Euclidean convex body is a measure
on the unit sphere which extends the notion of Gauss curvature to non-smooth
bodies. Alexandrov's problem consists in finding a convex body with given
curvature measure. In Euclidean space, A.D. Alexandrov gave a necessary and
sufficient condition on the measure for this problem to have a solution. In
this paper, we address Alexandrov's problem for convex bodies in the hyperbolic
space H m+1. After defining the Gauss curvature measure of an arbitrary
hyperbolic convex body, we completely solve Alexandrov's problem in this
setting. Contrary to the Euclidean case, we also prove the uniqueness of such a
convex body. The methods for proving existence and uniqueness of the solution
to this problem are both new.
