The dual Minkowski problem for positive indices

TL;DR

This paper establishes stability and existence results for the dual Minkowski problem with positive indices, showing solutions exist and are unique when the measure's density is close to uniform, using stability analysis and geometric measure theory.

Contribution

It provides the first stability result for the dual curvature measure with near constant density and proves existence and uniqueness of solutions for the even dual Minkowski problem with positive indices.

Findings

Stability of dual curvature measure near constant density.

Existence and uniqueness of solutions for the dual Minkowski problem.

Solutions are valid when the measure's density is close to 1 in the $C^{eta}$ norm.

Abstract

We derive the stability result of the dual curvature measure with near constant density in the even case. As an application, the existence and uniqueness of solutions to the even dual Minkowski problem for positive indices in $\mathbb{R}^{n+1}$ are obtained with $n\geq 1$, provided the density of the given measure is close to 1 in the $C^{\alpha}$ norm with $\alpha\in (0,1)$.