  We prove that the Gromov-Hausdorff distance from the circle with its geodesic
metric to any simply connected geodesic space is never smaller than
$\frac{\pi}{4}$. We also prove that this bound is tight through the
construction of a simply connected geodesic space $\mathrm{E}$ which attains
the lower bound $\frac{\pi}{4}$. 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 $(X, d_X)$ and
$(Y, d_Y )$ has to be in order for $\pi_1(X)$ and $\pi_1(Y)$ to be isomorphic.
