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2019 On the dimension and smoothness of radial projections
Tuomas Orponen
Anal. PDE 12(5): 1273-1294 (2019). 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,K2 are nonempty Borel sets with dimHK>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 dimHE>0, and E does not lie on a line, then there exists a point in xE such that the radial projection πx(K) has Hausdorff dimension at least (dimHK)2. Applying the result with E=K gives the following corollary: if K2 is a Borel set which does not lie on a line, then the set of directions spanned by K has Hausdorff dimension at least (dimHK)2.

For the second result, let d2 and d1<s<d. Let μ be a compactly supported Radon measure in d with finite s-energy. I prove that the radial projections of μ are absolutely continuous with respect to d1 for every centre in d sptμ, outside an exceptional set of dimension at most 2(d1)s. In fact, for x outside an exceptional set as above, the proof shows that πxμLp(Sd1) for some p>1. The dimension bound on the exceptional set is sharp.

Citation

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Tuomas Orponen. "On the dimension and smoothness of radial projections." Anal. PDE 12 (5) 1273 - 1294, 2019. https://doi.org/10.2140/apde.2019.12.1273

Information

Received: 23 November 2017; Revised: 31 July 2018; Accepted: 16 September 2018; Published: 2019
First available in Project Euclid: 5 January 2019

zbMATH: 1405.28011
MathSciNet: MR3892404
Digital Object Identifier: 10.2140/apde.2019.12.1273

Subjects:
Primary: 28A80
Secondary: 28A78

Rights: Copyright © 2019 Mathematical Sciences Publishers

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