  For every nonempty compact convex subset $K$ of a normed linear space a
(unique) point $c_K \in K$, called the generalized Chebyshev center, is
distinguished. It is shown that $c_K$ is a common fixed point for the isometry
group of the metric space $K$. With use of the generalized Chebyshev centers,
the central measure $\mu_X$ of an arbitrary compact metric space $X$ is
defined. For a large class of compact metric spaces, including the interval
$[0,1]$ and all compact metric groups, another `central' measure is
distinguished, which turns out to coincide with the Lebesgue measure and the
Haar one for the interval and a compact metric group, respectively. An idea of
distinguishing infinitely many points forming a dense subset of an arbitrary
compact metric space is also presented.
