PART METRIC IN CONVEX SETS

Any convex set C without lines in a linear space L can be decomposed into disjoint convex subsets (called parts) in a way which generalizes the idea of Gleason parts for a function space or function algebra. A metric d (called part metric) can be defined on C in a purely geometric way such that the parts of C are the components in the c-topology. This paper treats the connection between the convex structure of C and the metric d. The situation is particularly interesting when C is closed with respect to a weak Hausdorff topology on L (defined by a duality between L and another linear space).Then C is characterized by the set C+ of all continuous affine functions F on L satisfying F(x) ≧ 0 for all x ∈ C. This allows us to define d in terms of the functions log F, F ∈ C+. Furthermore, d-completeness of C can be derived from the completeness of C in L. The “ convexity” of the metric d leads to the existence of a continuous selection function for lower semi-continuous mappings of a paracompact space into the nonempty enclosed convex supsets of one part of such a complete convex set C We apply this result and the study of thepart metric of the convex cone of positive Radon measures on a locally compact Hausdorfif space to the problem of selecting in a continuous way mutually absolutely continuousrepresenting measures for points in one part of a function space or function algebra. © 1969 by Pacific Journal of Mathematics.