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Transactions of the American Mathematical Society

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Curve packing and modulus estimates

Authors: Katrin Fässler and Tuomas Orponen
Journal: Trans. Amer. Math. Soc. 370 (2018), 4909-4926
MSC (2010): Primary 28A75; Secondary 31A15, 60CXX
Published electronically: February 28, 2018
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Abstract | References | Similar Articles | Additional Information

Abstract: A family of planar curves is called a Moser family if it contains an isometric copy of every rectifiable curve in $ \mathbb{R}^{2}$ of length one. The classical ``worm problem'' of L. Moser from 1966 asks for the least area covered by the curves in any Moser family. In 1979, J. M. Marstrand proved that the answer is not zero: the union of curves in a Moser family always has area at least $ c$ for some small absolute constant $ c > 0$. We strengthen Marstrand's result by showing that for $ p > 3$, the $ p$-modulus of a Moser family of curves is at least $ c_{p} > 0$.

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Additional Information

Katrin Fässler
Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, FI-40014 Jyväskylä, Finland
Address at time of publication: Department of Mathematics, University of Fribourg, CH-1700 Fribourg, Switzerland

Tuomas Orponen
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014 Helsinki, Finland

Received by editor(s): July 25, 2016
Received by editor(s) in revised form: October 13, 2016
Published electronically: February 28, 2018
Additional Notes: The first author was supported by the Academy of Finland through the grant Sub-Riemannian manifolds from a quasiconformal viewpoint, grant number 285159
The second author was supported by the Academy of Finland through the grant Restricted families of projections and connections to Kakeya type problems, grant number $274512$. The second author is also a member of the Finnish CoE in Analysis and Dynamics Research.
Article copyright: © Copyright 2018 American Mathematical Society

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