A note on Krein-Milman theorem

Hubert Przybycień

doi:10.56754/0719-0646.2703.653

Authors

Hubert Przybycień Faculty of Mathematics and Natural Sciences UJK, ul. Uniwersytecka 7, 25-406 Kielce, Poland. https://orcid.org/0000-0002-7125-6113

DOI:

https://doi.org/10.56754/0719-0646.2703.653

Keywords:

Convex sets, extremal points, Krein-Milman theorem

Abstract

In this note, we generalize a well-known theorem of Krein and Milman concerning the closed convex envelope of the extremal points of a compact convex set in a topological vector space to the case of an abstract convexity notion in a topological space with a convex structure defined on it. We also introduce some notions that will be useful to describe these generalizations.

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References

N. Bourbaki, Topological vector spaces. Chapters 1–5, ser. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1987, doi: 10.1007/978-3-642-61715-7.

M. Krein and D. Milman, “On extreme points of regular convex sets”, Studia Math., vol. 9, pp. 133–138, 1940, doi: 10.4064/sm-9-1-133-138.

D. Mil’man, “Characteristics of extremal points of regularly convex sets”, Doklady Akad. Nauk SSSR (N.S.), vol. 57, pp. 119–122, 1947.

H. Przybycień, “A note on continuity of convex functions”, Novi Sad Journal of Mathematics, doi: 10.30755/NSJOM.16194.