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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: 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 \[ \Gamma _x=\{u\in \Gamma : [x,u]\cap \Gamma =\{u\}\}.\] (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 $x$, \[ \dim _H (\Gamma _x)\leq \frac {1}{2}+\sqrt {\dim _H(\Gamma )-\frac {3}{4}}.\] 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.