On an anisotropic Minkowski problem

TL;DR

This paper addresses the anisotropic Minkowski problem by formulating it as a Monge-Ampère type equation and proves existence and uniqueness of solutions, advancing understanding in anisotropic convex geometry.

Contribution

It introduces a novel formulation of the anisotropic Minkowski problem as a Monge-Ampère equation and establishes the first rigorous proof of existence and uniqueness of solutions.

Findings

Existence of solutions to the anisotropic Minkowski problem.

Uniqueness of the solution under given conditions.

Affirmative resolution of the 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.