Central points and measures and dense subsets of compact metric spaces

TL;DR

This paper introduces the generalized Chebyshev center for compact convex subsets in normed spaces, defines a central measure for compact metric spaces, and discusses dense subsets, connecting these concepts to fixed points and classical measures.

Contribution

It presents a novel concept of the generalized Chebyshev center, defines a central measure for compact metric spaces, and explores dense subsets, linking geometric and measure-theoretic properties.

Findings

The generalized Chebyshev center is a unique fixed point for the isometry group.

A central measure coincides with Lebesgue and Haar measures in specific cases.

A method for identifying dense subsets in compact metric spaces is proposed.

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.