  Many geometric optimization problems can be reduced to finding points in
space (centers) minimizing an objective function which continuously depends on
the distances from the centers to given input points. Examples are $k$-Means,
Geometric $k$-Median/Center, Continuous Facility Location, $m$-Variance, etc.
We prove that, for any fixed $\varepsilon>0$, every set of $n$ input points in
fixed-dimensional space with the metric induced by any vector norm admits a set
of $O(n)$ candidate centers which can be computed in almost linear time and
which contains a $(1+\varepsilon)$-approximation of each point of space with
respect to the distances to all the input points. It gives a universal
approximation-preserving reduction of geometric center-based problems with
arbitrary continuity-type objective functions to their discrete versions where
the centers are selected from a fairly small set of candidates. The existence
of such a linear-size set of candidates is also shown for any metric space of
fixed doubling dimension.
