  We show that, given $k> 4$, $0<J<\min\{{1\over 4},{k-4\over 4}\}$, any point
in space of non-degenerate smooth K\"ahler potentials has a small neighborhood
with respect to $C^k$ norm, s.t. any two points in this neighborhood can be
connected by a geodesic of at least $C^{k-J}$ regularity.
