On the dimension and smoothness of radial projections

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

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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: University of Helsinki, Department of Mathematics and Statistics
Date: 2019
Language: eng
Number of pages: 22
Belongs to series: Analysis & PDE
ISSN: 1948-206X
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 <s <d. 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 Hd-1 for every centre in R-d\spt mu,, 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 pi(x#)mu is an element of E L-p (Sd-1) for some p > 1. The dimension bound on the exceptional set is sharp.
Subject: Hausdorff dimension
fractals
radial projections
visibility
HAUSDORFF DIMENSION
VISIBILITY
111 Mathematics
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