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Visible parts and dimensions

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Published 28 February 2003 • Published under licence by IOP Publishing Ltd
, , Citation Esa Järvenpää et al 2003 Nonlinearity 16 803 DOI 10.1088/0951-7715/16/3/302

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1 Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD), FIN-40014, Finland

2 Department of Mathematics, Phillips Exeter Academy, 20 Main St, Exeter, NH 03833-2460, USA

3 Department of Pure Mathematics, The Open University, Walton Hall, Milton Keynes, Bucks MK7 6AA, UK

Dates

  1. Received 29 May 2002
  2. Published

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0951-7715/16/3/803

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 Bbb 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.

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10.1088/0951-7715/16/3/302