Planar Bilipschitz Extension from Separated Nets

Abstract

We prove that every $L$-bilipschitz mapping ${\mathbb{Z}}^2\to {\mathbb{R}}^2$ can be extended to a $C(L)$-bilipschitz mapping ${\mathbb{R}}^2\to {\mathbb{R}}^2$ and provide a polynomial upper bound for $C(L)$. Moreover, we extend the result to every separated net in ${\mathbb{R}}^2$ instead of ${\mathbb{Z}}^2$, with the upper bound gaining a polynomial dependence on the separation and net constants associated to the given separated net. This answers an Oberwolfach question of Navas from 2015 and is also a positive solution of the two-dimensional form of a decades old open (in all dimensions at least two) problem due to Alestalo, Trotsenko and Väisälä.

arXiv commentAccepted in Journal of the London Mathematical Society. Minor revision following the referee's report

References in corpus (2)

• Some examples of non-rectifiable, repetitive Delone sets
• Optimal extension of Lipschitz embeddings in the plane