  We study the properties of sets $\Sigma$ having the minimal length
(one-dimensional Hausdorff measure) over the class of closed connected sets
$\Sigma \subset \mathbb{R}^2$ satisfying the inequality $\mbox{max}_{y \in M}
\mbox{dist}(y,\Sigma) \leq r$ for a given compact set $M \subset \mathbb{R}^2$
and some given $r > 0$. Such sets can be considered shortest possible pipelines
arriving at a distance at most $r$ to every point of $M$ which in this case is
considered as the set of customers of the pipeline.
  We prove the conjecture of Miranda, Paolini and Stepanov about the set of
minimizers for $M$ a circumference of radius $R>0$ for the case when $r <
R/4.98$. Moreover we show that when $M$ is a 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.
