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PreprintarXiv licencearXiv

Visibility problem in higher dimensions

  • Yuming He
  • Changbiao Jian
  • Yudong Liang
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Abstract

We show that an almost-everywhere upper bound b≥ 1 for the Hausdorff dimension of visible parts of all compact planar sets implies the upper bound d−2+b for visible parts of all compact sets K in ℝd, d≥3. Inserting the sharp planar bound b=3/2 of Orponen and Rutar \cite{OrponenRutar2026} gives dimH Vise(K)≤ d−1/2 for almost every e∈ Sd−1. This would answer their Question 1.9 sharply in all dimensions.

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Cite

BibTeXRIS
BibTeX
@misc{he2026visibility,
  title = {{Visibility problem in higher dimensions}},
  author = {Yuming He and Changbiao Jian and Yudong Liang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.35388},
  eprint = {2609.35388},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35388},
}
RIS
TY  - UNPB
TI  - Visibility problem in higher dimensions
AU  - Yuming He
AU  - Changbiao Jian
AU  - Yudong Liang
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.35388
UR  - https://arxiv.org/abs/2609.35388
AB  - We show that an almost-everywhere upper bound b≥ 1 for the Hausdorff dimension of visible parts of all compact planar sets implies the upper bound d−2+b for visible parts of all compact sets K in ℝ^d, d≥3. Inserting the sharp planar bound b=3/2 of Orponen and Rutar \cite{OrponenRutar2026} gives dim_H Vis_e(K)≤ d−1/2 for almost every e∈ S^d−1. This would answer their Question 1.9 sharply in all dimensions.
ER  -