The $L_p$ dual Minkowski problem for unbounded closed convex sets

Abstract

The central focus of this paper is the $L_p$ dual Minkowski problem for $C$-compatible sets, where $C$ is a pointed closed convex cone in ${\mathbb{R}}^n$ with nonempty interior. Such a problem deals with the characterization of the $(p,q)$-th dual curvature measure of a $C$-compatible set. It produces new Monge-Ampère equations for unbounded convex hypersurface, often defined over open domains and with non-positive unknown convex functions. Within the family of $C$-determined sets, the $L_p$ dual Minkowski problem is solved for $0\ne p\in \mathbb{R}$ and $q\in \mathbb{R}$; while it is solved for the range of $p\le 0$ and $p<q$ within the newly defined family of $(C,p,q)$-close sets. When $p\le q$, we also obtain some results regarding the uniqueness of solutions to the $L_p$ dual Minkowski problem for $C$-compatible sets.