#### Vol. 12, No. 5, 2019

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On the dimension and smoothness of radial projections

### Tuomas Orponen

Vol. 12 (2019), No. 5, 1273–1294
##### 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 ${\pi }_{x}\left(K\right)$ has Hausdorff dimension at least $\left({dim}_{H}K\right)∕2$. 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 $\left({dim}_{H}K\right)∕2$.

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

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##### Keywords
Hausdorff dimension, fractals, radial projections, visibility
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