Geometric bounds for Favard length
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- by Rosemarie Bongers
- Proc. Amer. Math. Soc. 147 (2019), 1447-1452
- DOI: https://doi.org/10.1090/proc/14358
- Published electronically: December 19, 2018
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Abstract:
Given a set in the plane, the average length of its projections over all directions is called Favard length. This quantity measures the size of a set and is closely related to metric and geometric properties of the set such as rectifiability, Hausdorff dimension, and analytic capacity. In this paper, we develop new geometric techniques for estimating Favard length. We will give a short geometrically motivated proof relating Hausdorff dimension to the decay rate of the Favard length of neighborhoods of a set. We will also show that the sequence of Favard lengths of the generations of a self-similar set is convex; this has direct applications to giving lower bounds on Favard length for various fractal sets.References
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Bibliographic Information
- Rosemarie Bongers
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
- Received by editor(s): April 12, 2018
- Published electronically: December 19, 2018
- Additional Notes: Research partially supported by NSF grant DMS-1056965.
- Communicated by: Alexander Iosevich
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 1447-1452
- MSC (2010): Primary 28A78, 28A80
- DOI: https://doi.org/10.1090/proc/14358
- MathSciNet review: 3910411