  The famous Christoffel problem is possibly the oldest problem of prescribed
curvatures for convex hypersurfaces in Euclidean space. Recently, this problem
has been naturally formulated in the context of uniformly $h$-convex
hypersurfaces in hyperbolic space by Espinar-G\'alvez-Mira.
  Surprisingly, Espinar-G\'alvez-Mira find that the Christoffel problem in
hyperbolic space is essentially equivalent to the Nirenberg-Kazdan-Warner
problem on prescribing scalar curvature on $\mathbb{S}^n$. This equivalence
opens a new door to study the Nirenberg-Kazdan-Warner problem.
  In this paper, we establish a existence of solutions to the Christoffel
problem in hyperbolic space by proving a full rank theorem. As a corollary, a
existence of solutions to the Nirenberg-Kazdan-Warner problem follows.
