Abstract
The set St(f1, f2, f3) of Steiner points is described for any three functions f1, f2, f3 in the space C[K] of real-valued continuous functions on a Hausdorff compact set K. The set St(f1, f2, f3) consists of all functions s ∈ C[K] such that the sum ‖f1 − s‖ + ‖f2 − s‖ + ‖f3 − s‖ is minimal. It is proved that the set St(f1, f2, f3) is not empty; the triples f1, f2, f3 having a unique Steiner point are described; a Lipschitz selection is presented for the mapping (f1, f2, f3) → St(f1, f2, f3). These results imply the description of all real two-dimensional Banach spaces possessing the following property: the sum ‖x1 − s‖ + x2 − s‖ + ‖x3 − s‖ is equal to the semiperimeter of the triangle x1x2x3 for any triple x1, x2, x3 and some of its Steiner point s = s(x1, x2, x3).
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