Bi-Lipschitz embedding metric triangles in the plane
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908628489863168 |
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| author | Luo, Xinyuan Romney, Matthew Tao, Alexandria L. |
| author_facet | Luo, Xinyuan Romney, Matthew Tao, Alexandria L. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_20019 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bi-Lipschitz embedding metric triangles in the plane Luo, Xinyuan Romney, Matthew Tao, Alexandria L. Metric Geometry 51F30 (Primary), 30L05 (Secondary) 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. |
| title | Bi-Lipschitz embedding metric triangles in the plane |
| topic | Metric Geometry 51F30 (Primary), 30L05 (Secondary) |
| url | https://arxiv.org/abs/2407.20019 |