Strong Convergence of Alternating Projections

In this paper, we provide a necessary and sufficient condition under which the method of alternating projections on Hadamard spaces converges strongly. This result is new even in the context of Hilbert spaces. In particular, we found the circumstance under which the iteration of a point by projections converges strongly and we answer partially the main question that motivated Bruck's paper (J Math Anal Appl 88:319-322, 1982). We apply this condition to generalize Prager's theorem for Hadamard manifolds and generalize Sakai's theorem for a larger class of the sequences with full measure with respect to Bernoulli measure. In particular, we answer to a long-standing open problem concerning the convergence of the successive projection method (Aleyner and Reich in J Convex Anal 16:633-640, 2009). Furthermore, we study the method of alternating projections for a nested decreasing sequence of convex sets on Hadamard manifolds, and we obtain an alternative proof of the convergence of the proximal point method.