Regularity of geodesics in the spaces of convex and plurisubharmonic functions

Abja, Soufian; Dinew, Sławomir

doi:10.1090/tran/8350

Regularity of geodesics in the spaces of convex and plurisubharmonic functions
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by Soufian Abja and Sławomir Dinew PDF

Trans. Amer. Math. Soc. 374 (2021), 3783-3800 Request permission

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 $C^{1,1}$ regularity. We construct examples which prove that the global $C^{1,1}$ 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.

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