On an anisotropic Minkowski problem

Abstract: In this paper, we study the anisotropic Minkowski problem. It is a problem of prescribing the anisotropic Gauss-Kronecker curvature for a closed strongly convex hypersurface in Euclidean space as a function on its anisotropic normals in relative or Minkowski geometry. We first formulate such problem to a Monge-Amp\'ere type equation on the anisotropic support function and then prove the existence and uniqueness of the admissible solution to such equation. In conclusion, we give an affirmative answer to the anisotropic Minkowski problem.