Multilinear generalized Radon transforms and point configurations
By Loukas Grafakos, Allan Greenleaf, Alex Iosevich, Eyvindur A. Palsson. 2012
55 citations recorded by the source corpus.
We study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case. These provide a general mechanism to study Falconer-type problems involving $(k+1)$-point configurations in geometric measure theory, with $k \ge 2$, including the distribution of simplices, volumes and angles determined by the points of fractal subsets $E \subset {\Bbb R}^d$, $d \ge 2$. If $T_k(E)$ denotes the set of noncongruent $(k+1)$-point configurations determined by $E$, we show that if the Hausdorff dimension of $E$ is greater than $d-\frac{d-1}{2k}$, then the ${k+1 \choose 2}$-dimensional Lebesgue measure of $T_k(E)$ is positive. This compliments previous work on the Falconer conjecture (\cite{Erd05} and the references there), as well as work on finite point configurations \cite{EHI11,GI10}. We also give applications to Erd\"os-type problems in discrete geometry and a fractal regular value theorem, providing a multilinear framework for the results in \cite{EIT11}.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Computer Vision and Pattern Recognition
View Paper: https://www.semanticscholar.org/paper/fe198cb91eb029f43e4568b0b4ed442a5c465fd4
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In Applied Mathematics
In Computational Theory and Mathematics
In Computer Vision and Pattern Recognition
Citations in the corpus
- A group-theoretic viewpoint on Erdos-Falconer problems and the Mattila integral (2013, methodology): “The exponent obtained in [8], [7] is sd,d = d−1/2+1/(2d).”