  In this note, we shall prove geodesic convexity of the space of K\"ahler
potentials on an ALE K\"ahler manifold. This extends earlier results in the
compact case proved in the fundamental work of X-X. Chen. We further prove the
boundedness from below of the Mabuchi energy, and give an alternative proof for
the uniqueness of scalar-flat metrics in the category of ALE K\"ahler manifold.
