Geodesic Convexity of Small Neighborhood in the Space of K\"ahler Potentials

We show that, given $k>4$, $0<J<\min \{\frac{1}{4},\frac{k-4}{4}\}$, 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.