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

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arxiv 2404.09804 v1 pith:MGLTS4MY submitted 2024-04-15 math.MG math.AP

Wen Ai , Yunlong Yang , Deping Ye This is my paper · ORCID
classification math.MGmath.AP
keywords dualproblemsetsconvexminkowskicompatiblemathbbclosed
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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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Cited by 4 Pith papers

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  1. The Gaussian Minkowski-type problems for $C$-pseudo-cones

    math.MG 2025-01 conditional novelty 7.0 of 10

    New existence and conditional uniqueness theorems for Gaussian Minkowski and log-Minkowski problems on C-pseudo-cones.

  2. The Gaussian Minkowski problem for epigraphs of convex functions

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    The Gaussian Minkowski problem is generalized to epigraphs of convex functions, and existence of convex functions realizing prescribed Gaussian moment measures is established under mild conditions.

  3. The Gaussian-Minkowski problem for $C$-pseudo-cones

    math.FA 2024-12 reject novelty 6.0 of 10

    For every finite measure on the polar directions of a pointed cone, there exists a C-pseudo-cone whose Gaussian surface area measure equals that measure, with Gaussian co-volume no more than half the cone's.

  4. Minkowski Problems for Geometric Measures

    math.MG 2025-02 unverdicted novelty 2.0 of 10

    A comprehensive survey that organizes the Minkowski problems of convex geometry into a unified framework based on geometric measures as differentials of global geometric invariants.

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