# Uniqueness of solutions to Lp-Christoffel-Minkowski problem for p<1

## TL;DR

This paper proves a uniqueness theorem for solutions to the $L_p$-Christoffel-Minkowski problem when p<1, addressing a fundamental issue in convex geometric analysis by introducing a new auxiliary function.

## Contribution

It establishes the first uniqueness result for the $L_p$-Christoffel-Minkowski problem for p<1, using novel auxiliary functions and techniques inspired by curvature flow analysis.

## Key findings

- Proved uniqueness of solutions for p<1
- Introduced a new auxiliary function Z
- Extended understanding of convex geometric analysis

## Abstract

$L_p$-Christoffel-Minkowski problem arises naturally in the $L_p$-Brunn-Minkowski theory. It connects both curvature measures and area measures of convex bodies and is a fundamental problem in convex geometric analysis. Since the lack of Firey's extension of Brunn-Minkowski inequality and constant rank theorem for $p<1$, the existence and uniqueness of $L_p$-Brunn-Minkowski problem are difficult problems. In this paper, we prove a uniqueness theorem for solutions to $L_p$-Christoffel-Minkowski problem with $p<1$ and constant prescribed data. Our proof is motivated by the idea of Brendle-Choi-Daskaspoulos's work on asymptotic behavior of flows by powers of the Gaussian curvature. One of the highlights of our arguments is that we introduce a new auxiliary function $Z$ which is the key to our proof.