Curve packing and modulus estimates
Fässler, K., & Orponen, T. (2018). Curve packing and modulus estimates. Transactions of the American Mathematical Society, 370(7), 4909-4926. https://doi.org/10.1090/tran/7175
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Transactions of the American Mathematical SocietyDate
2018Copyright
© 2018 American Mathematical Society
A family of planar curves is called a Moser family if it contains an isometric
copy of every rectifiable curve in 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 has always 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 cp > 0.
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