Illinois Journal of Mathematics

On the Krein–Milman–Ky Fan theorem for convex compact metrizable sets

Mohammed Bachir

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We extend the extension by Ky Fan of the Krein–Milman theorem. The $\Phi $-extreme points of a $\Phi $-convex compact metrizable space are replaced by the $\Phi $-exposed points in the statement of Ky Fan theorem. Our main results are based on new variational principles which are of independent interest. Several applications will be given.


Due to computer-generated errors that were introduced in the typesetting stage, this article has been reprinted in its entirety in the Illinois Journal of Mathematics (Volume 62:1–4, 2018, pp. 1-24). The publisher apologizes for any inconvenience to readers.

Article information

Illinois J. Math., Volume 61, Number 1-2 (2017), 1-24.

Received: 14 July 2016
Revised: 2 September 2017
First available in Project Euclid: 3 March 2018

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Zentralblatt MATH identifier

Primary: 46B22: Radon-Nikodým, Kreĭn-Milman and related properties [See also 46G10] 46B20: Geometry and structure of normed linear spaces 49J50: Fréchet and Gateaux differentiability [See also 46G05, 58C20]


Bachir, Mohammed. On the Krein–Milman–Ky Fan theorem for convex compact metrizable sets. Illinois J. Math. 61 (2017), no. 1-2, 1--24. doi:10.1215/ijm/1520046206.

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  • Reprinted with corrections: Mohammed Bachir. On the Krein–Milman–Ky Fan theorem for convex compact metrizable sets. Illinois Journal of Mathematics, Volume 62:1–4, 2018, pp. 1-24.