The $L_p$ Minkowski problem for $C$-close sets: existence and continuity

Wen Ai; Deping Ye; Baocheng Zhu

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arXiv · 2609.23962

The $L_p$ Minkowski problem for $C$-close sets: existence and continuity

Wen Ai·Deping Ye·Baocheng Zhu

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Abstract

Let $C$ be a pointed closed convex cone in $\mathbb{R}^n$ with nonempty interior, and let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^n$. The $L_p$ Minkowski problem for $C$-close sets is to determine, for a real number $p$ and a nonzero finite Borel measure $μ$ defined on $Ω_{C^\circ}=S^{n-1}\cap \mathrm{int} C^{\circ}$, whether there exists a $C$-close set $\mathds{A}$ such that $μ$ is the $L_p$ surface area measure of $\mathds{A}$. In this paper, we will solve the problem for $p\in (0,1)$ and for $μ$ being a nonzero finite Borel measure on $Ω_{C^\circ}$. Moreover, we establish the continuity of solutions to the $L_p$ Minkowski problem for $p\in [0, 1]$ in several settings.

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Wen Ai, Deping Ye, Baocheng Zhu. 2026-09-21. The $L_p$ Minkowski problem for $C$-close sets: existence and continuity. https://arxiv.org/abs/2609.23962

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