Monge-Amp\`ere measures on balanced polyhedral spaces
Algebraic Geometry
2026-03-10 v1 Combinatorics
Abstract
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--Amp\`ere measures, first associated with piecewise affine functions, and then we extend it to polyhedrally plurisubharmonic functions. We investigate polyhedral Monge--Amp\`ere 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--Amp\`ere equation.
Cite
@article{arxiv.2603.08664,
title = {Monge-Amp\`ere measures on balanced polyhedral spaces},
author = {Ana María Botero and Enrica Mazzon and Léonard Pille-Schneider},
journal= {arXiv preprint arXiv:2603.08664},
year = {2026}
}