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Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika, 2011, Number 6, Pages 26–31
(Mi vmumm732)
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This article is cited in 6 scientific papers (total in 6 papers)
Mathematics
Steiner points in the space of continuous functions
B. B. Bednov Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
The set $\mathrm{St}(f_1,f_2,f_3)$ of Steiner points is described for any three functions $f_1,f_2,f_3$ in the space $C[\mathcal{K}]$ of real-valued continuous functions on a Hausdorff compact set $\mathcal{K}$. $\mathrm{St}(f_1,f_2,f_3)$ consists of all functions
$s\in C[\mathcal{K}]$ such that the sum $\|f_1-s\|+\|f_2-s\|+\|f_3-s\|$ is minimal. It is proved that the set $\mathrm{St}(f_1,f_2,f_3)$ is not empty; the triples $f_1,f_2,f_3$ having a unique Steiner point are described; a Lipschitz selection is presented for the mapping
$(f_1,f_2,f_3)\to\mathrm{St}(f_1,f_2,f_3)$. These results imply the description of all real two-dimensional Banach spaces possessing the following property: the sum $\|x_1-s\|+\|x_2-s\|+\|x_3-s\|$ is equal to the semiperimeter of triangle $x_1 x_2 x_3$ for any triple
$x_1,x_2,x_3$ and some of its Steiner point $s=s(x_1,x_2,x_3)$.
Key words:
Steiner point, space of continuous functions.
Received: 07.02.2011
Citation:
B. B. Bednov, “Steiner points in the space of continuous functions”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2011, no. 6, 26–31
Linking options:
https://www.mathnet.ru/eng/vmumm732 https://www.mathnet.ru/eng/vmumm/y2011/i6/p26
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