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 d-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 logF,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 d-closed convex subsets of one part of such a complete convex set C.
We apply this result and the study of the part metric of the convex cone
of positive Radon measures on a locally compact Hausdorff space to the
problem of selecting in a continuous way mutually absolutely continuous
representing measures for points in one part of a function space or function
algebra.
