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

TL;DR

This paper advances the understanding of geodesic regularity in spaces of convex and plurisubharmonic functions, establishing optimal regularity results and exploring the $C^{1,eta}$ regime, with applications to toric functions.

Contribution

It provides new optimal regularity results for geodesics between smooth convex functions and extends the theory to the $C^{1,eta}$ setting, including toric cases.

Findings

Optimal regularity for geodesics between smooth convex functions.

Regularity results in the $C^{1,eta}$ class.

Discussion of geodesic regularity in toric plurisubharmonic functions.

Abstract

In this note we continue our investigation of geodesics in the space of convex and plurisubharmonic functions. We show optimal regularity for geodesics joining two smooth strictly convex functions. We also investigate the regularity theory in the $C^{1,\alpha}$ realm. Finally we discuss the regularity of geodesics joining two toric strictly plurisubharmonic functions.