Sets of constant distance from a Jordan curve

Abstract

We study the $\epsilon$-level sets of the signed distance function to a planar Jordan curve $\Gamma$, and ask what properties of $\Gamma$ ensure that the $\epsilon$-level sets are Jordan curves, or uniform quasicircles, or uniform chord-arc curves for all sufficiently small $\epsilon$. Sufficient conditions are given in term of a scaled invariant parameter for measuring the local deviation of subarcs from their chords. The chordal conditions given are sharp.

Cited by in corpus (3)

• Quasisymmetric spheres over Jordan domains
• Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps
• Weak chord-arc curves and double-dome quasisymmetric spheres