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 (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.

$C-$Robin Functions and Applications

“…the last equality due to w = 0 on supp(dd c V * C,K ) 2 . Thus V * C,K = 0 a.e.-(dd c w) 2 and hence V * C,K ≤ w a.e.-(dd c w) 2 . By Proposition 2.2,…”

“…Much of the recent development of this C−pluripotential theory can be found in [9], [1] and [2]. One noticeable item lacking from these works is a constructive approach to finding natural concrete families of polynomials associated to K, C which recover V C,K .…”

Polynomials associated to non-convex bodies

“…Let R + = [0, ∞) and fix a convex body C ⊂ (R + ) d (C is compact, convex and C o = ∅). Associated with C we consider the finite-dimensional polynomial spaces As in [3], [4], [9], except for the case of C = C 0 defined in (1.12), we make the assumption throughout the entire paper on C that (the exponents j k need not be integers) and we use H C to define a generalization of L(C d ):…”

Polynomials associated to non-convex bodies

“…For a nonconstant polynomial p we define deg C (p) = min{n ∈ N : p ∈ Poly(nC)}. (1.3) As in [3], [4], [9], we make the assumption on C that Σ ⊂ kC for some k ∈ Z + .…”