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The Lp Minkowski problem for C-close sets: existence and continuity
Abstract
Let C be a pointed closed convex cone in ℝn with nonempty interior, and let Sn−1 denote the unit sphere in ℝn. The Lp Minkowski problem for C-close sets is to determine, for a real number p and a nonzero finite Borel measure μ defined on ΩC∘=Sn−1∩ int C∘, whether there exists a C-close set mathdsA such that μ is the Lp surface area measure of mathdsA. In this paper, we will solve the problem for p∈ (0,1) and for μ being a nonzero finite Borel measure on ΩC∘. Moreover, we establish the continuity of solutions to the Lp Minkowski problem for p∈ [0, 1] in several settings.
Related
BibTeX
@misc{ai2026minkowski,
title = {{The $L_p$ Minkowski problem for $C$-close sets: existence and continuity}},
author = {Wen Ai and Deping Ye and Baocheng Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23962},
eprint = {2609.23962},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23962},
}
RIS
TY - UNPB
TI - The L_p Minkowski problem for C-close sets: existence and continuity
AU - Wen Ai
AU - Deping Ye
AU - Baocheng Zhu
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.23962
UR - https://arxiv.org/abs/2609.23962
AB - Let C be a pointed closed convex cone in ℝ^n with nonempty interior, and let S^n−1 denote the unit sphere in 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^∘=S^n−1∩ int C^∘, whether there exists a C-close set mathdsA such that μ is the L_p surface area measure of mathdsA. In this paper, we will solve the problem for p∈ (0,1) and for μ being a nonzero finite Borel measure on Ω_C^∘. Moreover, we establish the continuity of solutions to the L_p Minkowski problem for p∈ [0, 1] in several settings.
ER -