Pluripotential Theory and Convex Bodies: A Siciak-Zaharjuta theorem

TL;DR

This paper extends pluripotential theory by defining a new class of functions associated with convex bodies, proving a Siciak-Zaharjuta type theorem in this setting, and addressing regularization challenges.

Contribution

It introduces the class $L_P$ of plurisubharmonic functions linked to convex bodies and proves a generalized Siciak-Zaharjuta theorem using a special regularization method.

Findings

$L_P$ is not closed under standard smoothing operations.

A continuous regularization preserves $L_P$ functions.

The weighted $P$-extremal function can be approximated using specific functions in $L_P$.

Abstract

We work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $({\bf R}^+)^d$. We define the {\it logarithmic indicator function} on ${\bf C}^d$: $$H_P(z):=\sup_{ J\in P} \log |z^{ J}|:=\sup_{ J\in P} \log[|z_1|^{ j_1}\cdots |z_d|^{ j_d}]$$ and an associated class of plurisubharmonic (psh) functions: $$L_P:=\{u\in PSH({\bf C}^d): u(z)- H_P(z) =0(1), \ |z| \to \infty \}.$$ 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 $$V_{P,K,Q}(z):=\sup \{u(z):u\in L_P, \ u\leq Q \ \hbox{on} \ K\}$$ 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 $\frac{1}{deg_P(p)}\log |p|$ (appropriately normalized).