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On a compact complex affine manifold with a constant coefficient K\"ahlermetric $\omega_0$, we introduce a concept: $(S,\omega_0)$-convexity and showthat $(S,\omega_0)$-convexity is preserved by geodesics in the space ofK\"ahler potentials. This implies that if two potentials are both strictly$(S,\omega_0)$-convex, then the metrics along the geodesic connecting them arenon-degenerate.

The Preservation of Convexity by Geodesics in the Space of K\"ahler Potentials on Complex Affine Manifolds