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

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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 \frac {\pi }{4} . We also prove that this bound is tight through the construction of a simply connected geodesic space E \mathrm {E} which attains the lower bound π 4 \frac {\pi }{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 , d X ) (X, d_X) and ( Y , d Y ) (Y, d_Y ) has to be in order for π 1 ( X ) \pi _1(X) and π 1 ( Y ) \pi _1(Y) to be isomorphic.