The traveling salesman theorem for Jordan curves

Christopher J. Bishop

doi:10.1016/j.aim.2022.108443

The traveling salesman theorem for Jordan curves

Christopher J. Bishop

Mathematics

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

If Γ⊂Rⁿ, n≥2, is a Jordan arc with endpoints z and w, we show that the arclength of Γ satisfies ℓ(Γ)−|z−w|≃∑Qβ_Γ²(Q)diam(Q), where the sum is over all dyadic cubes in Rⁿ and β_Γ(Q) is Peter Jones's β-number that measures the deviation of Γ from a straight line inside 3Q. This estimate sharpens previously known results by replacing an O(diam(Γ)) term by |z−w|. Applications of this improvement to the study of Weil-Petersson curves are described, and a new proof of Jones's traveling salesman theorem is given.

Original language	English
Article number	108443
Journal	Advances in Mathematics
Volume	404
DOIs	https://doi.org/10.1016/j.aim.2022.108443
State	Published - Aug 6 2022

Keywords

Arclength
Integral geometry
Rectifiable sets
Traveling salesman theorem
Weil-Petersson class

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10.1016/j.aim.2022.108443

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title = "The traveling salesman theorem for Jordan curves",

abstract = "If Γ⊂Rn, n≥2, is a Jordan arc with endpoints z and w, we show that the arclength of Γ satisfies ℓ(Γ)−|z−w|≃∑QβΓ2(Q)diam(Q), where the sum is over all dyadic cubes in Rn and βΓ(Q) is Peter Jones's β-number that measures the deviation of Γ from a straight line inside 3Q. This estimate sharpens previously known results by replacing an O(diam(Γ)) term by |z−w|. Applications of this improvement to the study of Weil-Petersson curves are described, and a new proof of Jones's traveling salesman theorem is given.",

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N2 - If Γ⊂Rn, n≥2, is a Jordan arc with endpoints z and w, we show that the arclength of Γ satisfies ℓ(Γ)−|z−w|≃∑QβΓ2(Q)diam(Q), where the sum is over all dyadic cubes in Rn and βΓ(Q) is Peter Jones's β-number that measures the deviation of Γ from a straight line inside 3Q. This estimate sharpens previously known results by replacing an O(diam(Γ)) term by |z−w|. Applications of this improvement to the study of Weil-Petersson curves are described, and a new proof of Jones's traveling salesman theorem is given.

AB - If Γ⊂Rn, n≥2, is a Jordan arc with endpoints z and w, we show that the arclength of Γ satisfies ℓ(Γ)−|z−w|≃∑QβΓ2(Q)diam(Q), where the sum is over all dyadic cubes in Rn and βΓ(Q) is Peter Jones's β-number that measures the deviation of Γ from a straight line inside 3Q. This estimate sharpens previously known results by replacing an O(diam(Γ)) term by |z−w|. Applications of this improvement to the study of Weil-Petersson curves are described, and a new proof of Jones's traveling salesman theorem is given.

KW - Arclength

KW - Integral geometry

KW - Rectifiable sets

KW - Traveling salesman theorem

KW - Weil-Petersson class

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DO - 10.1016/j.aim.2022.108443

M3 - Article

AN - SCOPUS:85132660103

SN - 0001-8708

VL - 404

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 108443

ER -

The traveling salesman theorem for Jordan curves

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Keywords

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