IUMJ

Title: On an anisotropic Minkowski problem

Authors: Chao Xia

Issue: Volume 62 (2013), Issue 5, 1399-1430

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 $\mathbb{R}^{n+1}$ as a function on its anisotropic normals in relative or Minkowski geometry. We first reduce such a 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 an equation. In conclusion, we give an affirmative answer to the anisotropic Minkowski problem.