# Symmetrization and extension of planar bi-Lipschitz maps

## TL;DR

This paper proves that centrally symmetric bi-Lipschitz circle embeddings in the plane can be extended to the entire plane with controlled distortion, improving bounds for general embeddings and introducing a symmetrization technique.

## Contribution

It establishes a linear distortion extension for symmetric embeddings and improves bounds for general cases, introducing a new symmetrization method.

## Key findings

- Centrally symmetric embeddings extend with linear distortion bounds.
- General embeddings have combined linear and cubic distortion bounds.
- A new symmetrization technique for bi-Lipschitz maps is developed.

## Abstract

We show that every centrally symmetric bi-Lipschitz embedding of the circle into the plane can be extended to a global bi-Lipschitz map of the plane with linear bounds on the distortion. This answers a question of Daneri and Pratelli in the special case of centrally symmetric maps. For general bi-Lipschitz embeddings our distortion bound has a combination of linear and cubic growth, which improves on the prior results. The proof involves a symmetrization result for bi-Lipschitz maps which may be of independent interest.