Pluripotential theory associated with general bodies

Pluripotential theory associated with a convex body $P\subset (\mathbb R^+)^d$ is a recently developing field of study. In this paper, motivated by work of J. J. Callaghan [Ann. Polon. Math. 90 (2007), 21–35] on $\theta $-incomplete polynomials in $\mathbb C^d$, we build the initial foundation of a pluripotential theory associated with more general bodies, specifically, when $P$ is the set-theoretic difference of two convex bodies $P_1, P_2$. We prove an approximation result for $P$-extremal functions and a global domination principle in this setting.