Polyhedral direct sums of Banach spaces, and generalized centers of finite sets

A Banach space $X$ is said to satisfy (GC) if the set $E_f(a)$ of minimizers of the function $X\ni x\mapsto f(\|x-a_1\|,\ldots,\|x-a_n\|)$ is nonempty for each integer $n\ge1$, each $a\in X^n$ and each continuous nondecreasing coercive real-valued function $f$ on $\R^n_+$. We study stability of certain polyhedrality properties under making direct sums, in order to be able to use results from a paper by Fonf, Lindenstrauss and the author to show that if $X$ satisfies (GC) and an appropriate polyhedrality property then the function space $C_b(T,X)$ satisfies (GC) for every topological space $T$. This generalizes the author's result from 1997, proved for finite dimensional polyhedral spaces $X$. Moreover, under more restrictive conditions on $X$ and $f$, the mappings $E_f(\cdot)$ on $C(K,X)^n$ ($n\ge1$) are continuous in the Hausdorff metric for each compact $K$.