The Preservation of Convexity by Geodesics in the Space of Kähler Potentials on Complex Affine Manifolds

On a compact complex affine manifold with a constant coefficient Kähler metric ${\omega }_0$, we introduce a concept: $(S,{\omega }_0)$-convexity and show that $(S,{\omega }_0)$-convexity is preserved by geodesics in the space of Kähler potentials. This implies that if two potentials are both strictly $(S,{\omega }_0)$-convex, then the metrics along the geodesic connecting them are non-degenerate.