Järvenpää, Esa; Järvenpää, Maarit; MacManus, Paul and O'Neil, Toby C.
|DOI (Digital Object Identifier) Link:||http://doi.org/10.1088/0951-7715/16/3/302|
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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 Article|
|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/Department:||Mathematics, Computing and Technology > Mathematics and Statistics
Mathematics, Computing and Technology
|Depositing User:||Toby O'Neil|
|Date Deposited:||13 Nov 2006|
|Last Modified:||14 Jan 2016 15:46|
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