  The Fermat-Steiner problem consists in finding all points in a metric space
$Y$ such that the sum of distances from each of them to the points from some
fixed finite subset of $Y$ is minimal. This problem is investigated for the
metric space $Y=H(X)$ of compact subsets of a metric space $X$, endowed with
the Hausdorff distance. For the case of a proper metric space $X$ a description
of all compacts $K\in H(X)$ which the minimum is attained at is obtained. In
particular, the Steiner minimal trees for three-element boundaries are
described. We also construct an example of a regular triangle in $H(R^2)$, such
that all its shortest trees have no "natural" symmetry.
