The Hausdorff dimension of visible sets of planar continua

Abstract

For a compact set Γ ⊂ R 2 \Gamma \subset \mathbb {R}^2 and a point x x , we define the visible part of Γ \Gamma from x x to be the set \[ Γ x = { u ∈ Γ : [ x , u ] ∩ Γ = { u } } . \Gamma _x=\{u\in \Gamma : [x,u]\cap \Gamma =\{u\}\}. \] (Here [ x , u ] [x,u] denotes the closed line segment joining x x to u u .) In this paper, we use energies to show that if Γ \Gamma is a compact connected set of Hausdorff dimension greater than one, then for (Lebesgue) almost every point x ∈ R 2 x\in \mathbb {R}^2 , the Hausdorff dimension of Γ x \Gamma _x is strictly less than the Hausdorff dimension of Γ \Gamma . In fact, for almost every x x , \[ dim H ⁡ ( Γ x ) ≤ 1 2 + dim H ⁡ ( Γ ) − 3 4 . \dim _H (\Gamma _x)\leq \frac {1}{2}+\sqrt {\dim _H(\Gamma )-\frac {3}{4}}. \] We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension greater than σ + 1 2 + dim H ⁡ ( Γ ) − 3 4 \sigma +\frac {1}{2}+\sqrt {\dim _H (\Gamma )-\frac {3}{4}} for some σ > 0 \sigma >0 .