2016/01/15 by Alexandr Ivanov, Ivanov, Alexandr, Alexandr Tropin +3 · 1 citation
Mathematics · #49Q20 #51Fxx #53C23 #58Exx #FOS: Mathematics #Metric Geometry (math.MG) #math.MG #msc:49Q20 #msc:51Fxx #msc:53C23 #msc:58Exx
paper · pdf · doi:10.48550/arxiv.1601.03852
15 pages, 5 Figures
arxiv created 2016/01/15 · arxiv updated 2016/01/18
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∈ 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(R2), such that all its shortest trees have no "natural" symmetry.