Monge-Ampère measures on balanced polyhedral spaces

We study classes of convex functions on balanced polyhedral spaces and establish various structural properties, including a compactness theorem for polyhedrally plurisubharmonic functions. Using tropical intersection theory, we construct Monge–Ampère measures, first associated with piecewise affine functions, and then we extend it to polyhedrally plurisubharmonic functions. We investigate polyhedral Monge–Ampère equations on balanced polyhedral spaces via a variational approach, providing sufficient conditions for the existence of solutions as well as explicit counterexamples. Finally, we relate our framework to non-archimedean pluripotential theory and explore its connection with the non-archimedean Monge–Ampère equation.