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 $σ\in S^{1}$, let $\mathrm{Vis}_σ(K) \subset K$ be the visible part of $K$ in direction $σ$. We prove that $\operatorname{dim}_{\mathrm{H}} \mathrm{Vis}_σ(K) \leq \tfrac{3}{2}$ for $\mathcal{H}^{1}$ almost every $σ\in S^{1}$. This is sharp: we construct a compact set $K \subset \mathbb{R}^{2}$ such that $\operatorname{dim}_{\mathrm{H}} \mathrm{Vis}_σ(K)\geq \tfrac{3}{2}$ for all $σ\in S^1$.