The mapping taking three points of a Banach space to their Steiner point

A mapping $\mathrm{St}$ sending any three points $a, b, c$ of a Banach space $X$ into a set $\mathrm{St}(a, b, c)$ of their medians and a corresponding operator $P_D$ of metric projection of a space $X \times X \times X$ onto its diagonal subspace $D=\{(x, x, x) \colon x \in X\}$, $P_D(a, b, c)=\{(s, s, s) \colon s \in \mathrm{St}(a, b, c)\}$, are considered. The linearity coefficient of arbitrary selection from $P_D$ is estimated, depending on different properties of the space $X$. As a corollary, estimates for the Lipschitz constant of arbitrary selection from the mapping $\mathrm{St}$ are obtained.