Bi-Lipschitz embedding metric triangles in the plane

TL;DR

This paper establishes the optimal distortion bound for bi-Lipschitz embedding of metric triangles into the Euclidean plane, providing detailed examples and demonstrating the limitations of such embeddings for quadrilaterals.

Contribution

It proves the sharp distortion bound for tripodal embeddings of metric triangles and analyzes specific examples to demonstrate bounds and limitations.

Findings

Sharp distortion bound of 4√(7/3) for tripodal embeddings

Examples confirming the bound's sharpness

Embedding does not extend to metric quadrilaterals

Abstract

A metric polygon is a metric space comprised of a finite number of closed intervals joined cyclically. The second-named author and Ntalampekos recently found a method to bi-Lipschitz embed an arbitrary metric triangle in the Euclidean plane with uniformly bounded distortion, which we call here the tripodal embedding. In this paper, we prove the sharp distortion bound $4\sqrt{7/3}$ for the tripodal embedding. We also give a detailed analysis of four representative examples of metric triangles: the intrinsic circle, the three-petal rose, tripods and the twisted heart. In particular, our examples show the sharpness of the tripodal embedding distortion bound and give a lower bound for the optimal distortion bound in general. Finally, we show the triangle embedding theorem does not generalize to metric quadrilaterals by giving a family of examples of metric quadrilaterals that are not bi-Lipschitz embeddable in the plane with uniform distortion.