Pluripotential Theory and Convex Bodies: A Siciak-Zaharjuta theorem

We work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $({\mathbf{R}}^{+}{)}^d$. We define the {\it logarithmic indicator function} on ${\mathbf{C}}^d$: $$H_P(z):=\underset{J\in P}{\sup }\log \mathrm{\mid }{z}^J\mathrm{\mid }:=\underset{J\in P}{\sup }\log [\mathrm{\mid }{z}_1{\mathrm{\mid }}^{j_1}\cdots \mathrm{\mid }{z}_d{\mathrm{\mid }}^{j_d}]$$ and an associated class of plurisubharmonic (psh) functions: $$L_P:=\{u\in PSH({\mathbf{C}}^d):u(z)-H_P(z)=0(1),\mathrm{\mid }z\mathrm{\mid }\to \mathrm{\infty }\}\mathrm{.}$$ We first show that $L_P$ is not closed under standard smoothing operations. However, utilizing a continuous regularization due to Ferrier which preserves $L_P$, we prove a general Siciak-Zaharjuta type-result in our $P-$setting: the weighted $P-$extremal function $$V_{P,K,Q}(z):=\sup \{u(z):u\in L_P,u\le Q\text{on}K\}$$ associated to a compact set $K$ and an admissible weight $Q$ on $K$ can be obtained using the subclass of $L_P$ arising from functions of the form $\frac{1}{de{g}_P(p)}\log \mathrm{\mid }p\mathrm{\mid }$ (appropriately normalized).