# The Hausdorff dimension of visible sets of planar continua.

## Toby C O'Neil

*Transactions of the AMS* **359** Number 11 (2007), 5141-5170

## 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.

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**Created:***21 May 2003*

**Modified:***22 January 2009*