Visible parts and dimensions
- 28 February 2003
- journal article
- Published by IOP Publishing in Nonlinearity
- Vol. 16 (3), 803-818
- https://doi.org/10.1088/0951-7715/16/3/302
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 )t oa only intersects A at a (here PK denotes orthogonal 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 dimen- sion 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.This publication has 9 references indexed in Scilit:
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