A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem

TL;DR

This paper introduces a constant rank theorem for linear elliptic equations on the sphere, enabling solutions to the mixed Christoffel problem to be interpreted as support functions of convex bodies, with applications in geometric analysis.

Contribution

It develops a novel constant rank theorem for elliptic PDEs on the sphere, facilitating geometric interpretation of solutions to the mixed Christoffel problem.

Findings

Established sufficient conditions for solving the mixed Christoffel problem.

Proved the constant rank property ensures solutions are support functions of convex bodies.

Applied the theorem to guarantee geometric solutions in convex geometry.

Abstract

We study the mixed Christoffel problem for $C^{2,+}$ convex bodies providing sufficient conditions for its solution. Key to our approach is a constant rank theorem, following the approach developed in \cite{Guan-Ma-2003} to address the Christoffel problem, in order to ensure that the solution to a related second order linear PDE on the sphere is indeed geometric, that is, it is the support functions of a $C^{2,+}$ convex body.