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In this note we investigate the regularity of geodesics in the space ofconvex and plurisubharmonic functions. In the real setting we prove (optimal)local C^{1,1} regularity. We construct examples which prove that the globalC^{1,1} regularity fails both in the real and complex case in contrast to theK\"ahler manifold setting. Finally we show a necessary and sufficientconditions for existence of a smooth geodesic between two smooth strictlyconvex functions.

Regularity of geodesics in the spaces of convex and plurisubharmonic functions