Fermat-Steiner Problem in the Metric Space of Compact Sets endowed with Hausdorff Distance

摘要

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 somefixed finite subset of $Y$ is minimal. This problem is investigated for themetric space $Y=H(X)$ of compact subsets of a metric space $X$, endowed withthe Hausdorff distance. For the case of a proper metric space $X$ a descriptionof all compacts $K\in H(X)$ which the minimum is attained at is obtained. Inparticular, the Steiner minimal trees for three-element boundaries aredescribed. We also construct an example of a regular triangle in $H(R^2)$, suchthat all its shortest trees have no "natural" symmetry.