Gromov-Hausdorff distances from simply connected geodesic spaces to the circle

We prove that the Gromov-Hausdorff distance from the circle with its geodesic metric to any simply connected geodesic space is never smaller than π/4. We also prove that this bound is tight through the construction of a simply connected geodesic space E which attains the lower bound π/4. We deduce the first statement from a general result that we also establish which gives conditions on how small the Gromov-Hausdorff distance between two geodesic metric spaces (X, dX) and (Y, dY ) has to be in order for π₁(X) and π₁(Y) to be isomorphic.