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arXiv · 2404.09804

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

Wen Ai·Yunlong Yang·Deping Ye
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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-Amp\`{e}re 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 $0\neq p\in \mathbb{R}$ and $q\in \mathbb{R}$; while it is solved for the range of $p\leq 0$ and $p<q$ within the newly defined family of $(C, p, q)$-close sets. When $p\leq q$, we also obtain some results regarding the uniqueness of solutions to the $L_p$ dual Minkowski problem for $C$-compatible sets.

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Wen Ai, Yunlong Yang, Deping Ye. 2024-04-15. The $L_p$ dual Minkowski problem for unbounded closed convex sets. https://arxiv.org/abs/2404.09804

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