Pluripotential theory and convex bodies: large deviation principle

Abstract

We continue the study in [2] in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $(\mathbb{R}^{+})^d$. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms of $P$-pluripotential-theoretic notions. As an important preliminary step, we first give an existence proof for the solution of a Monge–Ampère equation in an appropriate finite energy class. This is achieved using a variational approach.