The Hausdorff dimension of the visible sets of planar continua

O'Neil, Toby (2007). The Hausdorff dimension of the visible sets of planar continua. Transactions of the American Mathematical Society, 359(11) pp. 5141–5170.

URL: http://www.ams.org/tran/2007-359-11/S0002-9947-07-...

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 $\sigma > 0$ .