Pluripotential theory and convex bodies: large deviation principle

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We continue the study in [2] in the setting of weighted pluripotential theory arising from polynomials associated to a convex body P in (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-Amp\`ere equation in an appropriate finite energy class. This is achieved using a variational approach.