Pluripotential Theory and Convex Bodies

TL;DR

This paper extends pluripotential theory to convex bodies in real space, adapting complex geometric methods to analyze polynomial spaces not necessarily linked to line bundles, and recovers analogous asymptotic results.

Contribution

It generalizes pluripotential theory results from complex geometry to polynomial spaces associated with convex bodies in real space, broadening the scope of the theory.

Findings

Established asymptotic behavior of polynomial spaces related to convex bodies.

Extended complex geometric methods to real polynomial settings.

Recovered key results of pluripotential theory in a new context.

Abstract

In their seminal paper, Berman and Boucksom exploited ideas from complex geometry to analyze asymptotics of spaces of holomorphic sections of tensor powers of certain line bundles $L$ over compact, complex manifolds as the power grows. This yielded results on weighted polynomial spaces in weighted pluripotential theory in $\Bbb{C}^d$. Here, motivated from Bayraktar's recent paper, we work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body in $(\Bbb{R}^+)^d$. These classes of polynomials need not occur as sections of tensor powers of a line bundle $L$ over a compact, complex manifold. We follow the approach in Berman and Boucksom's work to recover analogous results.