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 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 orginal 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 dimension of the visible part.
We also investigate some examples of planar sets with Hausdorff dimension larger than 1. In particular, we prove for quasi-circles in the plane that all visible parts have Hausdorff dimension equal to 1.
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t.c.oneil@open.ac.ukCreated:24th May 2002
Modified:27th May 2003