We study mainly the class (GC) of all real Banach spaces $X$ such that the set $E_f(a)$ of the minimizers of the function X\ni x\mapsto f(\normx-a_1,\ldots,\normx-a_N) is nonempty whenever $N$ is a positive integer, $a\in X^N$, and $f$ is a continuous monotone coercive function on $[0,+\infty[^N$. For particular choices of $f$, the set $E_f(a)$ coincides with the set of Chebyshev centers of the set $\seta_ii=1,\ldots,N$ or with the set of its medians. The class (GC) is stable under making $c_0$-, $\ell^p$- and similar sums. Under some geometric conditions on $X$, the function spaces $C_b(T,X)$ or $L^p(\mu,X)$ belong to (GC). One of the main tools is a theorem which asserts that, in the definition of the class (GC), one can restrict himself to the functions $f$ of the type $f(\ks_1,\ldots,\ks_N)=\max \ro_i\ks_i$ ($\ro_i>0$).