We study the properties of sets Sigma having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets Sigma subset of R-2 satisfying the inequality max(y is an element of M) dist (y, Sigma) = r for a given compact set M subset of R-2 and some given r 0. Such sets play the role of shortest possible pipelines arriving at a distance at most r to every point of M, where M is the set of customers of the pipeline. We describe the set of minimizers for M a circumference of radius R 0 for the case when r R/4.98, thus proving the conjecture of Miranda, Paolini and Stepanov for this particular case. Moreover we show that when M is the boundary of a smooth convex set with minimal radius of curvature R, then every minimizer Sigma has similar structure for r R/5. Additionaly, we prove a similar statement for local minimizers.
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