The $L_p$-dual Christoffel-Minkowski problem for the case $p\geq q$

TL;DR

The paper tackles the L_p-dual Christoffel-Minkowski problem for p >= q by formulating a Hessian-type equation on the sphere and developing a constant rank framework to preserve convexity along a continuity path. For p > q, it proves existence and uniqueness of strictly spherical convex solutions via sharp a priori estimates and a continuity method. When p = q > 1, it handles the critical case through epsilon-regularization, obtaining uniform estimates and passing to a limit to obtain a unique solution up to dilation, with a uniquely determined scaling gamma. The core contributions include the constant rank theorem under suitable phi conditions and comprehensive C^0–C^2 estimates that enable existence and uniqueness results in both regimes, enriching the nonhomogeneous and density-fulfilling variants of the classical Christoffel-Minkowski problem. The results advance the dual Brunn-Minkowski theory by confirming solvability for a broad class of L_p-dual problems and clarifying the role of convexity preservation in the continuous deformation approaches.

Abstract

In this paper, we consider a class of Hessian equations associated to the $L_p$-dual Christoffel-Minkowski problem for the case $p\geq q$. By combining the tools of constant rank theorem, the a priori estimates and the continuity method, we obtain the existence and uniqueness for strictly spherical convex solutions to the $L_p$-dual Christoffel-Minkowski problem.

Theorems & Definitions (22)

• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Definition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5