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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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