Pluripotential Theory and Convex Bodies: Large Deviation Principle

TL;DR

This paper extends pluripotential theory related to convex bodies by establishing a large deviation principle and proving existence of solutions to a Monge-Ampère equation using variational methods.

Contribution

It introduces a large deviation principle in weighted pluripotential theory associated with convex bodies and provides an existence proof for a Monge-Ampère equation in this context.

Findings

Established a large deviation principle with a specific rate function.

Proved existence of solutions to a Monge-Ampère equation in a finite energy class.

Used a variational approach to solve the Monge-Ampère equation.

Abstract

We continue the study in a previous work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $({\bf R}^+)^d$. 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.