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 ∋ x ↦ f ( ‖ x − a 1 ‖ , … , ‖ x − a n ‖ ) is nonempty for each integer n ⩾ 1 , each a ∈ 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 authorʼs result from 1997, proved for finite-dimensional polyhedral spaces X. Moreover, under more restrictive conditions on X and f, the mappings E f ( ⋅ ) on C ( K , X ) n ( n ⩾ 1 ) are continuous in the Hausdorff metric for each compact K.