# The Hausdorff dimension of visible sets of planar continua.

## Abstract

For a compact set $$K\subset\mathbb{R}^2$$ and a point $$x$$, we define the visible part of $$K$$ from $$x$$ to be the set

$$K_x=\{u\in K : [x,u]\cap K=\{u\}\}.$$

(Here $$[x,u]$$ denotes the line segment joining $$x$$ to $$u$$.)

In this paper, we use energies to show that if $$K$$ is a compact connected set of Hausdorff dimension at least one, then for (Lebesgue) almost every point $$x\in\mathbb{R}^2$$,

$$\dim_H (K_x)\leq \frac{1}{2}+\sqrt{\dim_H (K)-\frac{3}{4}}.$$

We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension larger than $$\sigma+\frac{1}{2}+\sqrt{\dim_H (K)-\frac{3}{4}}$$ for $$\sigma>0$$.

Both [PDF, 320k] and [PS, 840k] versions of the preprint are available.