The Open UniversitySkip to content

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.

Full text available as:
PDF (Not Set) - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (347Kb)
Google Scholar: Look up in Google Scholar


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 $\sigma > 0$.

Item Type: Journal Article
ISSN: 0002-9947
Keywords: Hausdorff dimension; connected; visible sets
Academic Unit/Department: Mathematics, Computing and Technology > Mathematics and Statistics
Mathematics, Computing and Technology
Item ID: 2141
Depositing User: Toby O'Neil
Date Deposited: 15 Aug 2007
Last Modified: 25 Jul 2016 10:26
Share this page:

Download history for this item

These details should be considered as only a guide to the number of downloads performed manually. Algorithmic methods have been applied in an attempt to remove automated downloads from the displayed statistics but no guarantee can be made as to the accuracy of the figures.

▼ Automated document suggestions from open access sources

Actions (login may be required)

Policies | Disclaimer

© The Open University   + 44 (0)870 333 4340