Abstract
If Γ⊂Rn, n≥2, is a Jordan arc with endpoints z and w, we show that the arclength of Γ satisfiesℓ(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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