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Rectifiable sets and the traveling salesman problem. (English) Zbl 0731.30018
The author shows that a bounded set K (\(\subset {\mathbb{C}})\) is contained in a rectifiable curve if and only if \[ \sum \{\frac{\omega (Q)}{\ell (Q)}\}^ 2 \ell (Q)<\infty. \] Here the summation is taken over all dyadic squares Q, \(\ell (Q)\) denotes the sidelength of \({\mathbb{Q}}\) and \(\omega\) (Q) is the width of an infinite strip \(S_ Q\) with smallest possible width such that \(S_ Q\supset K\cap 3Q\). This assertion is very deep and applicable to study various problems concerning harmonic measure and the Cauchy integral on curves.
Reviewer: T.Murai (Nagoya)

MSC:
30C85 Capacity and harmonic measure in the complex plane
90C35 Programming involving graphs or networks
90C27 Combinatorial optimization
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References:
[1] Bishop, C.J., Jones, P.W.: Harmonic measure and arclength. Ann. Math. (to appear) · Zbl 0726.30019
[2] Calder?n, A.P.: Cauchy integrals on Lipschitz curves and related operators. Proc. Natl. Acad. Sci. U.S.A.74, 1324-1327 (1977) · Zbl 0373.44003 · doi:10.1073/pnas.74.4.1324
[3] Coifman, R.R., McIntosh, A., Meyer, Y.: L’int?grale de Cauchy d?finit un op?rateur born? sur L2 pour les courbes Lipschitziennes. Ann. Math.116, 361-368 (1982) · Zbl 0497.42012 · doi:10.2307/2007065
[4] Falconer, K.J.: The geometry of fractal sets, Cambridge University Press, 1985 · Zbl 0587.28004
[5] Federer, H.: Geometric measure theory. Berlin Heidelberg New York: Springer 1969 · Zbl 0176.00801
[6] Garnett, J.B.: Analytic capacity and measure (Lect. Notes Math. vol. 297) Berlin Heidelberg New York: Springer 1972 · Zbl 0253.30014
[7] Garnett, J.B.: Bounded analytic functions. Academic Press, 1981 · Zbl 0469.30024
[8] Jerison, D.S., Kenig, C.E.: Hardy spaces,A ? , and singular integrals on chord-arc domains. Math. Scand.50, 221-248 (1982) · Zbl 0509.30025
[9] Jones, P.W.: Square functions, Cauchy integrals, analytic capacity, and harmonic measure. (Lect. Notes Math, vol. 1384, pp. 24-68) Berlin Heidelberg New York: Springer 1989 · Zbl 0675.30029
[10] Jones, P.W.: Lipschitz and bi-Lipschitz functions. Revista Ibero-Americana4, 155-121 (1988)
[11] Jones, P.W., Murai, T.: Positive analytic capacity but zero Buffon needle probability. Pac. J. Math.133, 99-114 (1988) · Zbl 0653.30016
[12] Koosis, P.: Introduction to H p spaces. Lond. Math. Soc., Lecture Note Series, vol. 40, 1980 · Zbl 0435.30001
[13] Lawler, E.L.: The Traveling Salesman Problem. New York: Wiley-Interscience, 1985 · Zbl 0563.90075
[14] Mattila, P.: Smooth maps, null-sets for integral geometric measure and analytic capacity. Ann. Math.123, 303-309 (1986) · Zbl 0589.28006 · doi:10.2307/1971273
[15] Murai, T.: A real variable method for the Cauchy transform and analytic capacity. (Lect. Notes Math., vol. 1307) Berlin Heidelberg New York: Springer 1988 · Zbl 0645.30016
[16] Pommerenke, Ch.: Univalent functions. Vanderhoeck and Ruprecht, G?ttingen, 1975
[17] Preparata, F.P., Shamos, M.I.: Computational Geometry. Berlin Heidelberg New York: Springer 1985 · Zbl 0575.68059
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