Central points and measures and dense subsets of compact metric spaces

Piotr Niemiec

Abstract


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.

Keywords


Chebyshev center; convex set; common fixed point; Kantorovich metric; pointed metric space; distinguishing a point

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