Let \(X\) be a real Banach space. \(X\) is said to satisfy \((GC)\) if for every \(n\) and for every real-valued continuous, nondecreasing coercive function \(f\) on \( [0,\infty)^n\), the set \(E_f(a)\) of minimizers of the function \(\phi(x) = f(\|x-a_1\|,\dots,\|x-a_n\|)\) is nonempty, where \(x \in X\) and \(a= (a_1,\dots,a_n) \in X^n\). When this happens, finite sets in \(X\) have classical Chebyshev centers. By a weak\(^\ast\)-compactness argument, any dual space or a space which is the range of a projection of norm one in a dual space is in the class \(GC\). There are examples of finite dimensional spaces \(X\) for which the space of vector-valued bounded continuous functions \(C_b(T,X)\) fails to be in this class. In this very interesting paper, the author exhibits new classes of spaces in the class \(GC\) such that the space of continuous functions into them is also in this class. This is achieved by defining a new type of product of finitely many Banach spaces called the polyhedral direct sum. A Banach space is said to be polyhedral if the unit ball of every finite dimensional subspace is a polytope. For a polyhedral space \(X\), under an additional geometric condition, the author shows that \(C_b(T,X)\) satisfies \((GC)\) whenever \(X\) does.