Pluripotential Theory and Convex Bodies: A Siciak-Zaharjuta theorem

Abstract

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 ${\mathbf{C}}^d$: $$H_P(z):=\underset{J\in P}{\sup }\log |z^J|:=\underset{J\in P}{\sup }\log [|z_1{|}^{j_1}\cdots |z_d{|}^{j_d}]$$ and an associated class of plurisubharmonic (psh) functions: $$L_P:=\{u\in PSH({\mathbf{C}}^d):u(z)-H_P(z)=0(1),|z|\to \infty \}.$$ 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 $$V_{P,K,Q}(z):=\sup \{u(z):u\in L_P,u\le QonK\}$$ 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 $\frac{1}{de{g}_P(p)}\log |p|$ (appropriately normalized).

Cited by in corpus (1)

• Polynomials associated to non-convex bodies