One-point location problems from a very general point of view

Given a nonempty set $A$ in a metric space $(X,d)$, we consider the problem of minimizing a function $\phi\colon X\to\R$ of the form \[ \phi(x)=f(\Delta_x)\,, \] where $f$ is a monotone functional on the set $\lip(A)$ of all nonnegative $1$-Lipschitz functions on $A$, and $\Delta_x\colon A\to\R_+$ is the ($1$-Lipschitz) function $\Delta_x(\cdot)=d(x,\cdot)$. The minimizers (if any) of $\phi$ over $X$ are called {\em $f$-centers} of the set $A$. \par We present an existence theorem for $f$-centers, based on compactness in the so called $ball$-topology and on the Fatou property of $f$. We discuss sufficient conditions for the assumptions being satisfied. The last section is devoted to significant particular cases: generalized integral medians, generalized Chebyshev centers, and ``generalized centers with neglect''''.