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

Abstract

Lp-Christoffel-Minkowski problem arises naturally in the Lp-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 Lp-Brunn-Minkowski problem are difficult problems. In this paper, we prove a uniqueness theorem for solutions to Lp-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.