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

\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.