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Extreme points in non-positive curvature. (English) Zbl 1359.52002

The author of this note shows that there exists a bounded complete CAT(0) space without extreme points. Moreover, one can arrange that \(X\) is compact Hausdorf for the convex topology and that every finite collection of points in \(X\) is contained in a finite Euclidean simplicial complex of dimension two. Alternatively, one can construct a CAT(-1) example with hyperbolic simplicial complexes of dimension two.

MSC:

52A05 Convex sets without dimension restrictions (aspects of convex geometry)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C70 Direct methods (\(G\)-spaces of Busemann, etc.)
54G20 Counterexamples in general topology
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