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 K\"ahler manifold setting. Finally we show a necessary and sufficient conditions for existence of a smooth geodesic between two smooth strictly convex functions.

中文翻译：

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凸函数和多次谐波函数空间中测地线的正则性

在本笔记中，我们研究了凸函数和多次谐波函数空间中测地线的规律性。在真实环境中，我们证明了（最优）局部 C^{1,1} 规律。我们构建的例子证明全局 C^{1,1} 正则性在真实和复杂的情况下与 K\"ahler 流形设置相比都失败了。最后，我们展示了之间存在光滑测地线的充分必要条件两个光滑的严格凸函数。