Plane sets allowing bilipschitz extensions

Abstract

We give a geometric characterization for a plane set $A\subset {\mathsf R}^2$ to have the following linear bilipschitz extension property: For $0\le \varepsilon \le \delta$, every $(1 + \varepsilon)$-bilipschitz map $f\colon A\to {\mathsf R}^2$ has a $(1 + C\varepsilon)$-bilipschitz extension to the whole plane ${\mathsf R}^2$.