Visibility problem in higher dimensions

Yuming He; Changbiao Jian; Yudong Liang

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arXiv · 2609.35388

Visibility problem in higher dimensions

Yuming He·Changbiao Jian·Yudong Liang

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Abstract

We show that an almost-everywhere upper bound $b\geq 1$ 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 $\mathbb{R}^d$, $d\geq3$. Inserting the sharp planar bound $b=3/2$ of Orponen and Rutar \cite{OrponenRutar2026} gives $\dim_H \mathrm{Vis}_e(K)\leq d-1/2$ for almost every $e\in S^{d-1}$. This would answer their Question 1.9 sharply in all dimensions.

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BibTeX RIS

Yuming He, Changbiao Jian, Yudong Liang. 2026-09-28. Visibility problem in higher dimensions. https://arxiv.org/abs/2609.35388

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