Regularity of geodesics in the spaces of convex and plurisubharmonic functions

Abstract

In this note we investigate the regularity of geodesics in the space of convex and plurisubharmonic functions. In the real setting we prove (optimal) local C 1 , 1 C^{1,1} regularity. We construct examples which prove that the global C 1 , 1 C^{1,1} regularity fails both in the real and complex case in contrast to the Kähler manifold setting. Finally we show a necessary and sufficient conditions for existence of a smooth geodesic between two smooth strictly convex functions.