Bi-Lipschitz embeddings of quasiconformal trees

TL;DR

The paper proves that quasiconformal trees, a special class of metric trees, can be embedded into Euclidean space with controlled distortion, depending only on their doubling and bounded turning properties.

Contribution

It establishes bi-Lipschitz embedding results for quasiconformal trees, answering an open question and linking geometric properties to Euclidean embeddings.

Findings

Quasiconformal trees bi-Lipschitz embed into Euclidean space.

Embedding dimension and constants depend only on doubling and bounded turning constants.

Provides a positive answer to an open question in metric geometry.

Abstract

A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. In this paper we show that every quasiconformal tree bi-Lipschitz embeds in some Euclidean space, with the ambient dimension and the bi-Lipschitz constant depending only on the doubling and bounded turning constants of the tree. This answers Question 1.6 in \cite{DV} (arXiv:2007.12297).