Structure of sets with nearly maximal Favard length
Chang, A. lan; Dąbrowski, Damian; Orponen, Tuomas; Villa, Michele (2024-05-17)
Mathematical Sciences Publishers
17.05.2024
Chang, A., Dąbrowski, D., Orponen, T., & Villa, M. (2024). Structure of sets with nearly maximal Favard length. Analysis & PDE, 17(4), 1473–1500. https://doi.org/10.2140/apde.2024.17.1473.
https://creativecommons.org/licenses/by/4.0/
© The Author(s) 2024. MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY).
https://creativecommons.org/licenses/by/4.0/
© The Author(s) 2024. MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY).
https://creativecommons.org/licenses/by/4.0/
Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:oulu-202408125315
https://urn.fi/URN:NBN:fi:oulu-202408125315
Tiivistelmä
Abstract
Let E⊂B(1)⊂R2 be an H1 measurable set with H1(E)<∞, and let L⊂R2 be a line segment with H1(L)=H1(E). It is not hard to see that Fav(E)≤Fav(L). We prove that in the case of near equality, that is,
Fav(E)≥Fav(L)−δ,
the set E can be covered by an ϵ-Lipschitz graph, up to a set of length ϵ. The dependence between ϵ and δ is polynomial: in fact, the conclusions hold with ϵ=Cδ1∕70 for an absolute constant C>0.
Let E⊂B(1)⊂R2 be an H1 measurable set with H1(E)<∞, and let L⊂R2 be a line segment with H1(L)=H1(E). It is not hard to see that Fav(E)≤Fav(L). We prove that in the case of near equality, that is,
Fav(E)≥Fav(L)−δ,
the set E can be covered by an ϵ-Lipschitz graph, up to a set of length ϵ. The dependence between ϵ and δ is polynomial: in fact, the conclusions hold with ϵ=Cδ1∕70 for an absolute constant C>0.
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