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            <title xml:lang="en">On the Krein-Milman-Ky Fan theorem for convex compact metrizable sets.</title>
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                <title xml:lang="en">On the Krein-Milman-Ky Fan theorem for convex compact metrizable sets.</title>
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                <term xml:lang="en">$\Phi$-convexity</term>
                <term xml:lang="en">exposed and $\Phi$-exposed points</term>
                <term xml:lang="en">variational principle.</term>
                <term xml:lang="en">Krein-Milman theorem</term>
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              <p>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.</p>
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