This is an announcement for the paper "Central points and measures and dense subsets of compact metric spaces" by Piotr Niemiec.

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.

Archive classification: math.FA

Mathematics Subject Classification: Primary 46S30, 47H10, Secondary 46A55, 46B50

Remarks: 13 pages

Submitted from: piotr.niemiec@uj.edu.pl

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1105.5706

or

http://arXiv.org/abs/1105.5706