Central points and measures and dense subsets of compact metric spaces

الأصل · EN

For every nonempty compact convex subset K of a normed linear space a (unique) point cₖ ∈ K, called the generalized Chebyshev center, is distinguished. It is shown that cₖ is a common fixed point for the isometry group of the metric space K. With use of the generalized Chebyshev centers, the central measure μₓ 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.