The Weighted Dual Minkowski Problem Under Group Symmetry

Abstract

The paper studies the weighted dual Minkowski problem for the weighted qth dual curvature measure of convex bodies in ℝⁿ first posed by Huang, Lutwak, Yang, and Zhang. Most of the results are in the G-invariant setting, i.e. when the convex bodies are invariant under a closed subgroup G of O(n). They generalize previous results on the existence of solutions to the dual Minkowski problem for origin-symmetric convex bodies. It is proved: (i) a full characterization of the G-invariant weighted dual curvature measure for q\in(0,1], (ii) a necessary and sufficient condition for the existence of solutions to the G-invariant weighted dual Minkowski problem for q\in(1,n), (iii) a sufficient condition for the G-invariant weighted dual curvature measure for q>1, and (iv) an extension of Henk-Pollehn's dual curvature measure concentration property.

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BibTeX RIS

BibTeX

@misc{boroczky2026weighted,
  title = {{The Weighted Dual Minkowski Problem Under Group Symmetry}},
  author = {Karoly J. Boroczky and Stephanie Mui and Gaoyong Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20047},
  eprint = {2609.20047},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20047},
}

RIS

TY  - UNPB
TI  - The Weighted Dual Minkowski Problem Under Group Symmetry
AU  - Karoly J. Boroczky
AU  - Stephanie Mui
AU  - Gaoyong Zhang
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.20047
UR  - https://arxiv.org/abs/2609.20047
AB  - The paper studies the weighted dual Minkowski problem for the weighted qth dual curvature measure of convex bodies in ℝ^n first posed by Huang, Lutwak, Yang, and Zhang. Most of the results are in the G-invariant setting, i.e. when the convex bodies are invariant under a closed subgroup G of O(n). They generalize previous results on the existence of solutions to the dual Minkowski problem for origin-symmetric convex bodies. It is proved: (i) a full characterization of the G-invariant weighted dual curvature measure for q\in(0,1], (ii) a necessary and sufficient condition for the existence of solutions to the G-invariant weighted dual Minkowski problem for q\in(1,n), (iii) a sufficient condition for the G-invariant weighted dual curvature measure for q>1, and (iv) an extension of Henk-Pollehn's dual curvature measure concentration property.
ER  -

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