Helly's Intersection Theorem on Manifolds of Nonpositive Curvature

We give a generalization of the classical Helly's theorem on intersection of convex sets in $\mathbb{R}^N$ 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