Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$

TL;DR

This paper establishes a $C^0$ estimate for the $L_p$ $q$th dual Minkowski problem on the sphere in certain parameter ranges, leading to uniqueness results for near isotropic solutions.

Contribution

It provides the first $C^0$ estimate for the $L_p$ dual Minkowski problem in $R^3$ under broad conditions, extending previous results and addressing cases where $p$ is in [0,1) and $q>2+p$.

Findings

Proves $C^0$ estimate for specified $p,q$ ranges.

Shows uniqueness of solutions near isotropic case when $q$ is close to 3.

Identifies limitations of the estimate when $p<-1$ and $q=3$.

Abstract

We prove the $C^0$ estimate for the $L_p$ $q$th dual Minkowski problem on $S^2$ under fairly general conditions; namely, when $p$ lies in [0,1) and $q>2+p$, and the $L_p$ $q$th dual curvarture is bounded and bounded away from zero. We note that it is known that the analogous $C^0$ estimate does not hold if $p<-1$ and $q=3$. As a corollary of our $C^0$ estimate, we deduce the uniqueness of the solution of the near isotropic $q$th $L_p$ dual Minkowski problem on $S^2$ if $q$ is close to 3 and the $q$th $L_p$ dual curvature is Holder close to be the constant one function.