The $L_p$-Minkowski problem for $-n < p< 1$

TL;DR

This paper extends existence results for the $L_p$-Minkowski problem on the sphere for a range of p-values, providing new conditions especially for p in (0,1).

Contribution

It generalizes previous results to broader measures and offers an almost optimal condition for p in (0,1).

Findings

Extended existence results for $L_p$-Minkowski problem.

Provided an almost optimal condition for p in (0,1).

Broadened measure classes for the problem.

Abstract

Chou and Wang's existence result for the $L_p$-Minkowski problem on ${\mathbb S}^{n-1}$ for $p\in(-n,1)$ and an absolutely continuous measure $\mu$ is discussed and extended to more general measures. In particular, we provide an almost optimal sufficient condition for the case $p\in(0,1)$.