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. doi:10.1090/tran/7175
Published inTransactions of the American Mathematical Society
© 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.