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

## TL;DR

This paper studies the regularity of geodesics in spaces of convex and plurisubharmonic functions, establishing local regularity results, providing counterexamples for global regularity, and characterizing conditions for smooth geodesics.

## Contribution

It proves optimal local C^{1,1} regularity in the real setting, constructs counterexamples for global regularity, and characterizes when smooth geodesics exist between smooth convex functions.

## Key findings

- Proved local C^{1,1} regularity for geodesics in the real setting.
- Constructed examples showing failure of global C^{1,1} regularity.
- Identified necessary and sufficient conditions for smooth geodesics between smooth convex 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} 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.