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Transactions of the American Mathematical Society

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The Hausdorff dimension of visible sets of planar continua

Author: Toby C. O'Neil
Journal: Trans. Amer. Math. Soc. 359 (2007), 5141-5170
MSC (2000): Primary 28A80; Secondary 28A78, 31A15.
Published electronically: June 4, 2007
MathSciNet review: 2327025
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Abstract | References | Similar Articles | Additional Information

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

$\displaystyle \Gamma_x=\{u\in \Gamma: [x,u]\cap \Gamma=\{u\}\}.$

(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 greater than one, then for (Lebesgue) almost every point $ x\in\mathbb{R}^2$, the Hausdorff dimension of $ \Gamma_x$ is strictly less than the Hausdorff dimension of $ \Gamma$. In fact, for almost every $ x$,

$\displaystyle \dim_H (\Gamma_x)\leq \frac{1}{2}+\sqrt{\dim_H(\Gamma)-\frac{3}{4}}.$

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

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Additional Information

Toby C. O'Neil
Affiliation: Faculty of Mathematics and Computing, The Open University, Walton Hall, Milton Keynes, MK7 6AA, United Kingdom

Keywords: Visible sets, Hausdorff dimension
Received by editor(s): November 6, 2003
Published electronically: June 4, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.