Visibility problem in the plane

Abstract

We disprove the visibility conjecture in the plane and prove the sharp upper bound for the almost-sure dimension of visible parts. Precisely, let $K\subset {\mathbb{R}}^2$ be a compact set. For $\sigma \in S^1$, let ${\mathrm{Vis}}_{\sigma }(K)\subset K$ be the visible part of $K$ in direction $\sigma$. We prove that ${\dim }_{\mathrm{H}}{\mathrm{Vis}}_{\sigma }(K)\le \frac{3}{2}$ for ${\mathcal{H}}^1$ almost every $\sigma \in S^1$. This is sharp: we construct a compact set $K\subset {\mathbb{R}}^2$ such that ${\dim }_{\mathrm{H}}{\mathrm{Vis}}_{\sigma }(K)\ge \frac{3}{2}$ for all $\sigma \in S^1$.

50 pages, 4 figures. v2: Title changed. Combines the v1 upper bound with the corresponding lower bound due to the first author from arXiv:2606.29839