  This paper is a survey of plurisubharmonic theory where the usual polynomial
ring is replaced by a polynomial ring $\mathcal P^S(\mathbb C^n)$ where the
$m$-th degree polynomials have exponents restricted to $mS$, where $S\subseteq
\mathbb R^n_+$ is compact, convex and $0\in S$. We assume no other conditions
on $S$ as these are necessary for $\mathcal P^S(\mathbb C^n)$ to be a graded
polynomial ring. We study the relationship between $\mathcal P^S(\mathbb C^n)$
and the class $\mathcal L^S(\mathbb C^n)$ of global plurisubharmonic functions
where the growth is determined by the logarithmic supporting function of $S$.
We present properties of their respective weighted extremal functions $\Phi_{K,
q}^S$ and $V_{K, q}^S$ in connection with properties of $S$.
