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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 $$Y=H(X)$$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 $$K\in H(X)$$K∈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 $$H(\mathbb {R}^2)$$H(R2), such that all its shortest trees have no “natural” symmetry.

Fermat–Steiner problem in the metric space of compact sets endowed with Hausdorff distance