We study the properties of sets S having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets σ R² satisfying the inequality maxy2M dist (y; S) = r for a given compact set M R² 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 S has similar structure for r < R=5. Additionaly, we prove a similar statement for local minimizers.