Curve packing and modulus estimates

Fässler, Katrin
Department of Mathematics, University of Fribourg, Switzerland  Department of Mathematics and Statistics, University of Jyväskylä, Finland

Orponen, Tuomas
Department of Mathematics and Statistics, University of Jyväskylä, Finland
Published in:
 Transactions of the American Mathematical Society.  2018, vol. 370, no. 7, p. 4909–4926
English
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$

Faculty
 Faculté des sciences et de médecine

Department
 Département de Mathématiques

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Mathematics

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https://folia.unifr.ch/unifr/documents/306397
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