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

A Banach space X is said to satisfy (GC) if the set Ef(a) of minimizers of the function X∋x↦f(‖x−a1‖,…,‖x−an‖) is nonempty for each integer n⩾1, each a∈Xn and each continuous nondecreasing coercive real-valued function f on . 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 Cb(T,X) satisfies (GC) for every topological space T. This generalizes the authorʼs result from 1997, proved for finite-dimensional polyhedral spaces X. Moreover, under more restrictive conditions on X and f, the mappings Ef(⋅) on C(K,X)n (n⩾1) are continuous in the Hausdorff metric for each compact K.