  The Gauss Image problem is a generalization to the question originally posed
by Aleksandrov, who studied the existence of the convex body with prescribed
Aleksandrov's integral curvature. A simple discrete case of the Gauss Image
Problem can be formulated as follows: given a finite set of directions in
$\mathbb R^n$ and the same number of unit vectors, does there exist a convex
polytope in $\mathbb R^n$ containing the origin in its interior with vertices
at given directions such that each normal cone at the vertex contains exactly
one of the given vectors in its interior?
  We pose a combinatorial problem, called the Assignment Problem, for discrete
measures. It is shown that the discrete Gauss Image Problem and the Assignment
Problem are equivalent. We establish a geometric condition for measures which
solves the Assignment Problem and, hence, the Gauss Image Problem. We also show
that generically, the Assignment Problem has a solution. The proper
reformulation for the uniqueness questions is also addressed and analyzed. The
work establishes interesting connections of the Discrete Gauss Image problem to
Hall's marriage theorem and transportation polytopes.
