### Central points and measures and dense subsets of compact metric spaces

#### 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.

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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