Abstract
A sufficient condition for the existence of a continuous selector of representative measure, concentrated at the extreme points of a convex metrizable compactum, is considered. A necessary condition for the existence of such a selector is deduced. An example is given of a convex compactum with a closed set of extreme points, for which no continuous selector exists.
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H. Bauer and H. S. Bear, “The part metric in convex sets,” Pac. J. Math.,30, 15–34 (1969).
E. M. Alfsen, Compact Convex Sets and Boundary Integrals, Springer-Verlag, Berlin-New York (1971).
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Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 897–906, December, 1977.
In conclusion, the author expresses deep gratitude to A. M. Vershik for the formulation of the problem and for help in the preparation of this paper.
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Makarov, N.N. Continuous selector of representative measures and the space of faces of a convex compactum. Mathematical Notes of the Academy of Sciences of the USSR 22, 991–996 (1977). https://doi.org/10.1007/BF01099570
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DOI: https://doi.org/10.1007/BF01099570