On the dimension and smoothness of radial projections

Orponen, Tuomas

doi:10.2140/apde.2019.12.1273

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On the dimension and smoothness of radial projections

Tuomas Orponen

Vol. 12 (2019), No. 5, 1273–1294

DOI: 10.2140/apde.2019.12.1273

Abstract

This paper contains two results on the dimension and smoothness of radial projections of sets and measures in Euclidean spaces.

To introduce the first one, assume that $E, K \subset ℝ^{2}$ are nonempty Borel sets with ${dim}_{H} K > 0$ . Does the radial projection of $K$ to some point in $E$ have positive dimension? Not necessarily: $E$ can be zero-dimensional, or $E$ and $K$ can lie on a common line. I prove that these are the only obstructions: if ${dim}_{H} E > 0$ , and $E$ does not lie on a line, then there exists a point in $x \in E$ such that the radial projection $π_{x} (K)$ has Hausdorff dimension at least $({dim}_{H} K) ∕ 2$ . Applying the result with $E = K$ gives the following corollary: if $K \subset ℝ^{2}$ is a Borel set which does not lie on a line, then the set of directions spanned by $K$ has Hausdorff dimension at least $({dim}_{H} K) ∕ 2$ .

For the second result, let $d \geq 2$ and $d - 1 < s < d$ . Let $μ$ be a compactly supported Radon measure in $ℝ^{d}$ with finite $s$ -energy. I prove that the radial projections of $μ$ are absolutely continuous with respect to $ℋ^{d - 1}$ for every centre in $ℝ^{d} ∖ spt μ$ , outside an exceptional set of dimension at most $2 (d - 1) - s$ . In fact, for $x$ outside an exceptional set as above, the proof shows that $π_{x ♯} μ \in L^{p} (S^{d - 1})$ for some $p > 1$ . The dimension bound on the exceptional set is sharp.

Keywords

Hausdorff dimension, fractals, radial projections, visibility

Mathematical Subject Classification 2010

Primary: 28A80

Secondary: 28A78

Milestones

Received: 23 November 2017

Revised: 31 July 2018

Accepted: 16 September 2018

Published: 15 December 2018

Authors

Tuomas Orponen
Department of Mathematics and Statistics University of Helsinki Helsinki Finland

Vol. 12, No. 5, 2019