In the space of Kähler potentials, geodesic equations are homogeneous complex Monge–Ampère equations; the boundary value problems always have weak \(C^{1,1}\) solutions, but the Kähler metrics along the geodesic may be degenerate. In this paper, when the background manifold is a flat torus(or complex affine manifold), we introduce a concept: \((S, \omega _0)\) -convexity, which is a generalization of the usual convexity, and show that when two potentials are both strictly \((S, \omega _0)\) -convex, the Kähler metrics along the geodesic connecting them are non-degenerate. The result of this paper proves a conjecture of Guan and Phong (Math Ann 354(1):147–169, 2012) about the convexity of solutions to the homogeneous complex Monge–Ampère equation.
