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Bi-Lipschitz embedding metric triangles in the plane

Xinyuan Luo, Matthew Romney, Alexandria L. Tao

TL;DR

This work analyzes bi-Lipschitz embeddings of metric triangles in the plane, focusing on the tripodal embedding method and establishing a sharp distortion bound $D = 4\\sqrt{7/3}$ for all metric triangles. It develops a precise embedding construction via a tripod model and a case analysis, and it demonstrates sharpness through the twisted heart example, while also deriving a lower bound of $lip(\\triangle,\\mathbb{R}^2) \\ge 2$ from the three-petal rose. The authors further show that the analogous embedding result fails for metric quadrilaterals by constructing quadrilaterals that cannot be embedded in $\\mathbb{R}^2$ with uniform distortion, highlighting fundamental limits in 2D. Collectively, the results tightly characterize the 2D embeddability of metric triangles, provide explicit optimal bounds, and clarify the limitations when moving to larger polygonal classes.

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.

Bi-Lipschitz embedding metric triangles in the plane

TL;DR

This work analyzes bi-Lipschitz embeddings of metric triangles in the plane, focusing on the tripodal embedding method and establishing a sharp distortion bound for all metric triangles. It develops a precise embedding construction via a tripod model and a case analysis, and it demonstrates sharpness through the twisted heart example, while also deriving a lower bound of from the three-petal rose. The authors further show that the analogous embedding result fails for metric quadrilaterals by constructing quadrilaterals that cannot be embedded in with uniform distortion, highlighting fundamental limits in 2D. Collectively, the results tightly characterize the 2D embeddability of metric triangles, provide explicit optimal bounds, and clarify the limitations when moving to larger polygonal classes.

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 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.
Paper Structure (12 sections, 25 theorems, 100 equations, 5 figures)

This paper contains 12 sections, 25 theorems, 100 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Metric spaces and bi-Lipschitz maps
  4. Metric polygons
  5. Proof of optimal distortion bound
  6. Examples
  7. The intrinsic circle
  8. The three-petal rose
  9. Tripods
  10. The twisted heart
  11. Metric polygons
  12. Appendix

Key Result

Theorem 1.1

Every metric triangle embeds in $\mathbb{R}^2$ under the tripodal embedding with distortion at most $D = 4\sqrt{7/3} \approx 6.11$.

Figures (5)

  • Figure 1: Examples of metric polygons.
  • Figure 2: The five cases in the proof of \ref{['thm:embedding']}.
  • Figure 3:
  • Figure 4:
  • Figure 5: The metric polygons in \ref{['sec:quadrilaterals']}.

Theorems & Definitions (55)

  • Theorem 1.1
  • Proposition 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 3.1
  • ...and 45 more