# The Hausdorff dimension of the visible sets of planar continua

O'Neil, Toby (2007). The Hausdorff dimension of the visible sets of planar continua. Transactions of the American Mathematical Society, 359(11) pp. 5141–5170.

## Abstract

For a compact set and a point , we define the visible part of from to be the set

(Here denotes the closed line segment joining to .)

In this paper, we use energies to show that if is a compact connected set of Hausdorff dimension larger than one, then for (Lebesgue) almost every point , the Hausdorff dimension of is strictly
less than the Hausdorff dimension of . In fact, for almost every ,

We also give an estimate of the Hausdorff dimension of those points
where the visible set has dimension larger than for .