Visible parts and dimensions

Järvenpää, Esa; Järvenpää, Maarit; MacManus, Paul and O'Neil, Toby C. (2003). Visible parts and dimensions. Nonlinearity, 16(3) pp. 803–818.

DOI: 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) 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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