  We study the properties of sets $\Sigma$ which are the solutions of the
maximal distance minimizer problem, i.e. of sets having the minimal length
(one-dimensional Hausdorff measure) over the class of closed connected sets
$\Sigma \subset \mathbb{R}^n$ satisfying the inequality \[ max_{y \in M}
dist(y,\Sigma) \leq r \] for a given compact set $M \subset \mathbb{R}^n$ and
some given $r > 0$. Such sets can be considered as the shortest networks of
radiating Wi-Fi cables arriving to each customer (for the set $M$ of customers)
at a distance at most $r$.
  In this paper we prove that any maximal distance minimizer $\Sigma \subset
\mathbb{R}^n$ has at most $3$ tangent rays at each point and the angle between
any two tangent rays at the same point is at least $2\pi/3$. Moreover, in the
plane (for $n=2$) we show that the number of points with three tangent rays is
finite and every maximal distance minimizer is a finite union of simple curves
with one-sided tangents continuous from the corresponding side.
  All the results are proved for the more general class of local minimizers,
i.e. sets which are optimal under a perturbation of a neighbourhood of their
arbitrary point.

Metric Geometry