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

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