Regularity of geodesics in the spaces of convex and plurisubharmonic functions II

In this note we continue our investigation of geodesics in the space of convex and plurisubharmonic functions. We show optimal regularity for geodesics joining two smooth strictly convex functions. We also investigate the regularity theory in the $C^{1,\alpha }$ realm. Finally we discuss the regularity of geodesics joining two toric strictly plurisubharmonic functions.