Vol. 30, No. 1, 1969

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The part metric in convex sets

Heinz Bauer and Herbert Stanley Bear, Jr.

Vol. 30 (1969), No. 1, 15–33

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 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 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.

Mathematical Subject Classification
Primary: 46.01
Received: 13 November 1968
Published: 1 July 1969
Heinz Bauer
Herbert Stanley Bear, Jr.