CENTRAL POINTS AND MEASURES, AND DENSE SUBSETS OF COMPACT METRIC SPACES

For every nonempty compact convex subset K of a normed linear space a (unique) point c(K) is an element of 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.