We study the properties of sets Σ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets Σ ⊂ ℝ 2 satisfying the inequality max y ∈M dist ( y , Σ ) ≤ r for a given compact set M ⊂ 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 Σ has similar structure for r < R ∕ 5. Additionaly, we prove a similar statement for local minimizers.