The Discrete Gauss Image Problem

Abstract

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 R n and the same number of unit vectors, does there exist a convex polytope in 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.