# Visible parts and dimensions.

## 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 $$\mathbb{R}^n$$, 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.

A postscript (413K) and a pdf (239K) preprint of this paper are available.

If you have any queries about my work or would like a copy of this paper then please contact me at:

t.c.oneil@open.ac.uk

Created:24th May 2002
Modified:27th May 2003