The Hausdorff dimension of the visible sets of planar continua

Abstract

For a compact set $\Gamma\subset\Bbb{R}^2$ and a point $x$, we define the visible part of $\Gamma$ from $x$ to be the set $\Gamma_x = \{u \in\Gamma : [x, u] \cap\Gamma = \{u\}\}.$ (Here $[x, u]$ denotes the closed line segment joining $x$ to $u$.) In this paper, we use energies to show that if $\Gamma$ is a compact connected set of Hausdorff dimension larger than one, then for (Lebesgue) almost every point $x\in\Bbb{R}^2$, the Hausdorff dimension of $\Gamma_x$ is strictly less than the Hausdorff dimension of $\Gamma$. In fact, for almost every $x$, $\dim_H(\Gamma_x)\leq \frac{1}{2}+\sqrt{\dim_H(\Gamma){-}\frac{3}{4}}.$ We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension larger than $\sigma+\frac{1}{2}+\sqrt{{\dim_H}{(\Gamma)}{-}{\frac{3}{4}}}$ for $\sigma > 0$