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.
Fulltext:
PDF file (354 kB)