The Hausdorff dimension of visible sets of planar continua

For a compact set $\Gamma\subset\mathbb{R}^2$ 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 greater than one, then for (Lebesgue) almost every point $x\in\mathbb{R}^2$, 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 greater than $\sigma+\frac{1}{2}+\sqrt{\dim_H (\Gamma)-\frac{3}{4}}$ for some $\sigma>0$.