曲线填充和模估计
Curve packing and modulus estimates
摘要：
如果一个平面曲线族包含长度为1的$\mathbb{R}^{2}$中每一条可求长曲线的等距副本，则该平面曲线族称为Moser族。L. 1966年的Moser要求在任何Moser族中由曲线覆盖的最小面积。1979年，J.M.马斯特兰德证明，答案是不为零：工会的曲线在莫泽家庭始终面积至少$c$为一些小的绝对常数$c > 0$。我们加强Marstrand的结果表明，$p > 3$，$p$ -模的Moser族曲线至少是$c_{p} > 0$。
英文摘要：
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 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 $c_{p} > 0$.
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