On the Krein-Milman theorem in CAT(k) spaces

Category: Vol 34/2018 no. 3 Tags: CAT($\kappa $) spaces, extreme points, the Krein-Milman theorem

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Let $\kappa>0$ and $(X, \rho)$ be a complete CAT( $\kappa$ ) space whose diameter smaller than $\dfrac{\pi}{2\sqrt{\kappa}}$ . It is shown that if $K$ is a nonempty compact convex subset of $X,$ then $K$ is the closed convex hull of its set of extreme points. This is an extension of the Krein-Milman theorem to the general setting of CAT( $\kappa$ ) spaces.