Optimal extension of Lipschitz embeddings in the plane

TL;DR

This paper proves that bi-Lipschitz embeddings of the circle into the plane can be extended to the disk with controlled bi-Lipschitz constants, answering a previously open question.

Contribution

It establishes linear bounds for extending circle embeddings to the disk and plane, advancing understanding of Lipschitz extension problems.

Findings

Bi-Lipschitz circle embeddings extend to the disk with linear bounds.

Lipschitz circle embeddings extend to the entire plane with linear bounds.

Answers a question posed by Daneri and Pratelli.

Abstract

We prove that every bi-Lipschitz embedding of the unit circle into the plane can be extended to a bi-Lipschitz map of the unit disk with linear bounds on the constants involved. This answers a question raised by Daneri and Pratelli. Furthermore, every Lipschitz embedding of the circle extends to a Lipschitz homeomorphism of the plane, again with a linear bound on the constant.