  We establish a coarse version of the Cartan-Hadamard theorem, which states
that proper coarsely convex spaces are coarsely homotopy equivalent to the open
cones of their ideal boundaries. As an application, we show that such spaces
satisfy the coarse Baum-Connes conjecture. Combined with the result of
Osajda-Przytycki, it implies that systolic groups and locally finite systolic
complexes satisfy the coarse Baum-Connes conjecture.
