The $L_p$ dual Christoffel-Minkowski type problem for a class of Hessian quotient equations

TL;DR

This paper addresses a geometric problem involving Hessian quotient operators, establishing existence and uniqueness of convex solutions using inverse convexity properties and a full rank theorem.

Contribution

It introduces a new approach leveraging inverse convexity to solve the $L_p$ dual Christoffel-Minkowski problem for Hessian quotient equations.

Findings

Proved a full rank theorem under structural assumptions.

Established existence and uniqueness of convex solutions.

Utilized inverse convexity property of Hessian quotient operators.

Abstract

In this paper, we investigate an $L_p$ dual Christoffel-Minkowski type problem for the Hessian quotient operator $\frac{\sigma_{k}(\Lambda)}{\sigma_{l}(\Lambda)}$, where the operator $\Lambda$ has been widely studied in the literature. Exploiting the recently discovered ``inverse convexity'' property of this class of operators, we establish a full rank theorem under suitable structural assumptions. Together with a priori estimates, this result enables us to prove the existence and uniqueness of strictly spherically convex solutions to the above $L_p$ dual Christoffel-Minkowski type problem.