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Bauer simplex

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A non-empty compact convex subset of a locally convex space that is a Choquet simplex and such that the set of its extreme points is closed (cf. also Convex hull).

Bauer simplices are also characterized as the compact convex subsets such that every real-valued continuous function on can be extended to a (unique) continuous affine function on , or, equivalently, for which every point in is in the barycentre of a unique probability measure on supported by .

Such sets have been studied for the first time by H. Bauer [a3]. They were called Bauer simplices in [a1]. See [a1] for their relation with several aspects of convexity theory and potential theory.

More recently (1990s), new connections between them and some general problems in the approximation of continuous functions by positive operators and abstract degenerate elliptic-parabolic problems have been discovered (see, e.g., [a2]).

References

[a1] E.M. Alfsen, "Compact convex sets and boundary integrals" , Springer (1971)
[a2] F. Altomare, M. Campiti, "Korovkin type approximation theory and its applications" , W. de Gruyter (1994)
[a3] H. Bauer, "Schilowsche Rand und Dirichletsches Problem" Ann. Inst. Fourier , 11 (1961) pp. 89–136
How to Cite This Entry:
Bauer simplex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bauer_simplex&oldid=33240
This article was adapted from an original article by F. Altomare (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article