Järvenpää, Esa; Järvenpää, Maarit; MacManus, Paul and O'Neil, Toby C.
(2003).
| DOI (Digital Object Identifier) Link: | https://doi.org/10.1088/0951-7715/16/3/302 |
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| Google Scholar: | Look up in Google Scholar |
Abstract
We study the visible parts of subsets of n-dimensional Euclidean space: a point a of a compact set A is visible from an affine subspace K of Rn, if the line segment joining PK(a) to a only intersects A at a (here PK denotes projection onto K). The set of all such points visible from a given subspace K is called the visible part of A from K. We prove that if the Hausdorff dimension of a compact set is at most n−1, then the Hausdorff dimension of a visible part is almost surely equal to the Hausdorff dimension of the set. On the other hand, provided that the set has Hausdorff dimension larger than n − 1, we have the almost sure lower bound n − 1 for the Hausdorff dimensions of visible parts. We also investigate some examples of planar sets with Hausdorff dimension bigger than 1. In particular,we prove that for quasi-circles in the plane all visible parts have Hausdorff dimension equal to 1.
| Item Type: | Journal Item |
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| ISSN: | 1361-6544 |
| Extra Information: | Some of the symbols may not have transferred correctly into this bibliographic record and/or abstract. |
| Keywords: | Hausdorff dimension; visible sets |
| Academic Unit/School: | Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics Faculty of Science, Technology, Engineering and Mathematics (STEM) |
| Item ID: | 2138 |
| Depositing User: | Toby O'Neil |
| Date Deposited: | 13 Nov 2006 |
| Last Modified: | 04 Oct 2016 09:46 |
| URI: | http://oro.open.ac.uk/id/eprint/2138 |
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