  On a compact complex affine manifold with a constant coefficient K\"ahler
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
K\"ahler 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.
