Visibility problem in higher dimensions

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.