# On the dimension and smoothness of radial projections

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http://hdl.handle.net/10138/307024

#### Citation

Orponen , T 2019 , ' On the dimension and smoothness of radial projections ' , Analysis & PDE , vol. 12 , no. 5 , pp. 1273-1294 . https://doi.org/10.2140/apde.2019.12.1273

 Title: On the dimension and smoothness of radial projections Author: Orponen, Tuomas Contributor organization: Department of Mathematics and Statistics Date: 2019 Language: eng Number of pages: 22 Belongs to series: Analysis & PDE ISSN: 1948-206X DOI: https://doi.org/10.2140/apde.2019.12.1273 URI: http://hdl.handle.net/10138/307024 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 of 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 is an element of E such that the radial projection pi(x) (K) has Hausdorff dimension at least (dim(H)K)/2. Applying the result with E = K gives the following corollary: if K subset of 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 >= 2 and d - 1 1. The dimension bound on the exceptional set is sharp. Subject: Hausdorff dimension fractals radial projections visibility HAUSDORFF DIMENSION VISIBILITY 111 Mathematics Peer reviewed: Yes Usage restriction: openAccess Self-archived version: acceptedVersion
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