The Hausdorff dimension of visible sets of planar continua

Abstract:

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 $\sigma >0$.