  We introduce a particular class of unbounded closed convex sets of
$\R^{d+1}$, called F-convex sets (F stands for future). To define them, we use
the Minkowski bilinear form of signature $(+,...,+,-)$ instead of the usual
scalar product, and we ask the Gauss map to be a surjection onto the hyperbolic
space $\H^d$. Important examples are embeddings of the universal cover of
so-called globally hyperbolic maximal flat Lorentzian manifolds.
  Basic tools are first derived, similarly to the classical study of convex
bodies. For example, F-convex sets are determined by their support function,
which is defined on $\H^d$. Then the area measures of order $i$, $0\leq i\leq
d$ are defined. As in the convex bodies case, they are the coefficients of the
polynomial in $\epsilon$ which is the volume of an $\epsilon$ approximation of
the convex set. Here the area measures are defined with respect to the
  Lorentzian structure.
  Then we focus on the area measure of order one. Finding necessary and
sufficient conditions for a measure (here on $\H^d$) to be the first area
measure of a F-convex set is the Christoffel Problem. We derive many results
about this problem. If we restrict to "Fuchsian" F-convex set (those who are
invariant under linear isometries acting cocompactly on $\H^d$), then the
problem is totally solved, analogously to the case of convex bodies. In this
case the measure can be given on a compact hyperbolic manifold.
  Particular attention is given on the smooth and polyhedral cases. In those
cases, the Christoffel problem is equivalent to prescribing the mean radius of
curvature and the edge lengths respectively.
