ИСТИНА

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 $A$ of $Y$ is minimal. Such points are sometimes referred as geometric medians of $A$. 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 a surprising example of a quite symmetric regular triangle in $H(\R^2)$, such that all its shortest trees have no ``natural'' symmetry.