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 ∈ 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 µ 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. 2010 MSC: Primary 46S30, 47H10; Secondary 46A55, 46B50.