The growth rate of surface area measure for noncompact convex sets with prescribed asymptotic cone

TL;DR

This paper investigates the Minkowski problem for unbounded convex sets with a focus on the surface area measure growth rate, providing a complete solution in two dimensions and partial results in higher dimensions.

Contribution

It offers a complete characterization for the existence and uniqueness of solutions in 2D and extends partial results to higher dimensions for the Minkowski problem.

Findings

Complete solution for 2D case

Partial results for higher dimensions

Conditions for existence and uniqueness

Abstract

The Minkowski problem for a class of unbounded closed convex sets is considered. This is equivalent to a Monge-Amp\`ere equation on a bounded convex open domain with possibly non-integrable given data. A complete solution (necessary and sufficient condition for existence and uniqueness) in dimension 2 is presented. In higher dimensions, partial results are demonstrated.