Steiner points in the space of continuous functions

B. B. Bednov

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (2011), pp. 26-31

Résumé

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 $s=s(x_1,x_2,x_3)$.