The Hausdorff dimension of visible sets of planar continua.

Toby C O'Neil

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


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