Plurisubharmonic functions on Berkovich spaces and connectivity by curves

This is part of a joint project with Walter Gubler on pluripotential theory in non-Archimedean geometry. We introduce the notion of a "classically" plurisubharmonic function on a Berkovich space by requiring that all pullbacks to curves defined over any extension field be subharmonic, following complex geometry (the "classical" case). We argue that these form the largest reasonable class of psh functions. We prove that classically psh functions satisfy the local maximum principle using a "connectivity by curves" result: any two points in a connected analytic space can be joined by a chain of curves, defined over analytic field extensions.