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Steiner Points in the Space of Continuous Functions / Bednov B.B. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2011. № 6. P. 26-31 [Moscow Univ. Math. Bulletin. Vol. 66, N 6, 2011.]. 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).
Key words: Steiner point, space of continuous functions.
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