6533b86ffe1ef96bd12cdcdc
RESEARCH PRODUCT
Curve packing and modulus estimates
Tuomas OrponenKatrin FässlerKatrin Fässlersubject
General MathematicsTHIN SETModulusconformal modulus01 natural sciencesThin setpotential theoryCombinatoricsNull set010104 statistics & probabilityPlanarCIRCLESMathematics - Metric GeometryClassical Analysis and ODEs (math.CA)FOS: Mathematics111 Mathematics0101 mathematicsAbsolute constantMathematicsMoser familyApplied Mathematicsta111010102 general mathematicsMathematical analysisZero (complex analysis)Metric Geometry (math.MG)28A75 (Primary) 31A15 60CXX (Secondary)measure theoryMathematics - Classical Analysis and ODEsFamily of curvespotentiaaliteoriamittateoriaMEASURE ZEROcurve packing problemsdescription
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 $c_{p} > 0$.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2018-01-01 |