Geodesic
Helly's Intersection Theorem on Manifolds of Nonpositive Curvature
Journal of convex analysis, Tome 13 (2006) no. 3, pp. 785-798
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We give a generalization of the classical Helly's theorem on intersection of convex sets in RN for the case of manifolds of nonpositive curvature. In particular, we show that if any N+1 sets from a family of closed convex sets on N-dimensional Cartan-Hadamard manifold contain a common point, then all sets from this family contain a common point. 
@article{JCA_2006_13_3_JCA_2006_13_3_a19,
     author = {Y. S. Ledyaev and J. S. Treiman and Q. J. Zhu},
     title = {Helly's {Intersection} {Theorem} on {Manifolds} of {Nonpositive} {Curvature}},
     journal = {Journal of convex analysis},
     pages = {785--798},
     year = {2006},
     volume = {13},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JCA_2006_13_3_JCA_2006_13_3_a19/}
}
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Y. S. Ledyaev; J. S. Treiman; Q. J. Zhu. Helly's Intersection Theorem on Manifolds of Nonpositive Curvature. Journal of convex analysis, Tome 13 (2006) no. 3, pp. 785-798. http://geodesic.mathdoc.fr/item/JCA_2006_13_3_JCA_2006_13_3_a19/