We consider quasi-isometries in real continuous functions spaces and show that such a quasi-isometry can be well approximated by an affine surjective isometry.
On the other hand, we give an example of quasi-isometries of the unit ball
B
H
B_H
in a Hilbert space
H
H
that are far from any affine map of
H
H
and from any isometry of
B
H
B_H
.