Halpern’s Iteration in CAT(0) Spaces

Let X, d be a metric space and x, y ∈ X with l d x, y . A geodesic path from x to y is an isometry c : 0, l → X such that c 0 x and c l y. The image of a geodesic path is called a geodesic segment. A metric spaceX is a (uniquely) geodesic space if every two points ofX are joined by only one geodesic segment. A geodesic triangle x1, x2, x3 in a geodesic space X consists of three points x1, x2, x3 of X and three geodesic segments joining each pair of vertices. A comparison triangle of a geodesic triangle x1, x2, x3 is the triangle x1, x2, x3 : x1, x2, x3 in the Euclidean space R2 such that d xi, xj dR2 xi, xj for all i, j 1, 2, 3. A geodesic space X is a CAT(0) space if for each geodesic triangle : x1, x2, x3 in X and its comparison triangle : x1, x2, x3 in R2, the CAT(0) inequality