Uniqueness of solutions to a class of non-homogeneous curvature problems

TL;DR

This paper proves that the only smooth, convex, even solutions to certain isotropic curvature problems are spheres centered at the origin, resolving a longstanding question and establishing uniqueness results for related Minkowski problems.

Contribution

It establishes the uniqueness of origin-centered spherical solutions for a broad class of isotropic Minkowski-type problems, including the Orlicz-Minkowski and Lp-Gaussian-Minkowski problems.

Findings

Only even, smooth, convex solutions are spheres.

Spheres are the unique solutions to the considered curvature problems.

Results confirm conjectures about solution uniqueness in isotropic Minkowski problems.

Abstract

We show that the only even, smooth, convex solutions to a class of isotropic mixed Christoffel-Minkowski type problems are origin-centred spheres, which, in particular, answers a question of Firey 74 in the even isotropic case about kinematic measures. Employing the Heintze-Karcher inequality, we prove that the only smooth, strictly convex solutions to a large class of Minkowski type problems are origin-centred spheres. Immediate corollaries are the uniqueness of solutions to the isotropic Orlicz-Minkowski problem and the isotropic $L_p$-Gaussian-Minkowski problem when $p\geq 1$.