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 \(\mathbb{R}^n\), if the line segment joining \(P_K(a)\) to \(a\) only intersects \(A\) at \(a\) (here \(P_K\) 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