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 $({{\mathbb {R}}}^+)^d$. We define the logarithmic indicator function on ${{\mathbb {C}}}^d$:

$$\begin{aligned} H_P(z):=\sup _{ J\in P} \log |z^{ J}|:=\sup _{ J\in P} \log [|z_1|^{ j_1}\cdots |z_d|^{ j_d}] \end{aligned}$$

and an associated class of plurisubharmonic (psh) functions:

$$\begin{aligned} L_P:=\{u\in PSH({{\mathbb {C}}}^d): u(z)- H_P(z) =\mathcal {O}(1), \ |z| \rightarrow \infty \}. \end{aligned}$$

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

$$\begin{aligned} V_{P,K,Q}(z):=\sup \{u(z):u\in L_P, \ u\le Q \ \hbox {on} \ K\} \end{aligned}$$

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 $\frac{1}{\deg _P(p)}\log |p|$.

Appendix

We provide a version of the lemma from Ferrier [10] appropriate for our purposes to show $u_t$ in Proposition 3.1 is psh. For $\lambda >0$, we use the distance function $d_{\lambda }:{{\mathbb {C}}}^d \times {{\mathbb {C}}}\rightarrow [0,\infty )$ defined as $d_{\lambda }(z,w)=\lambda |z|+|w|$ (in our application, $t=1/\lambda$).

Lemma 5.1

Let $\delta :{{\mathbb {C}}}^d \rightarrow [0,\infty )$ be non-negative. For $\lambda >0$, define

$$\begin{aligned} {\widehat{\delta }}_{\lambda }(s):=\inf _{s'\in {{\mathbb {C}}}^d}[\delta (s')+\lambda |s'-s|]. \end{aligned}$$

Let

$$\begin{aligned} \Omega _1:=\{(s,t)\in {{\mathbb {C}}}^d \times {{\mathbb {C}}}: |t|< \delta (s)\}. \end{aligned}$$

Then

$$\begin{aligned} {\widehat{\delta }}_{\lambda }(s)= d_{\lambda }\bigl ( (s,0),({{\mathbb {C}}}^d \times {{\mathbb {C}}}) \setminus \Omega _1 \bigr ). \end{aligned}$$

(5.1)

Furthermore, if $\delta$ is lsc, then $\Omega _1$ is open. Moreover,

$$\begin{aligned} \Omega _1=\{(s,t): -\log \delta (s) + \log |t|<0\} \end{aligned}$$

so that if, in addition, $-\log \delta$ is psh in ${{\mathbb {C}}}^d$, then $\Omega _1$ is pseudoconvex in ${{\mathbb {C}}}^d \times {{\mathbb {C}}}$.

Proof

This is straightforward; first observe

$$\begin{aligned}&d_{\lambda }\bigl ( (s,0),({{\mathbb {C}}}^d \times {{\mathbb {C}}}) \setminus \Omega _1 \bigr )=\inf \{\lambda |s-s'|+ |t|:(s',t)\in ({{\mathbb {C}}}^d \times {{\mathbb {C}}}) \setminus \Omega _1 \} \\&\quad = \inf \{\lambda |s-s'|+ |t|:|t|\ge \delta (s') \} = \inf \{\lambda |s-s'|+ \delta (s'):s'\in {{\mathbb {C}}}^d \}={\widehat{\delta }}_{\lambda }(s). \end{aligned}$$

Next,

$$\begin{aligned} \Omega _1&:=\{(s,t)\in {{\mathbb {C}}}^d \times {{\mathbb {C}}}: |t|< \delta (s)\}\\&=\{(s,t)\in {{\mathbb {C}}}^d \times {{\mathbb {C}}}: -\log \delta (s) + \log |t|<0\}. \end{aligned}$$

$\square$

Corollary 5.2

Under the hypotheses of the lemma, if $-\log \delta$ is psh in ${{\mathbb {C}}}^d$ then $-\log {\widehat{\delta }}_{\lambda }$ is psh.

Proof

Since $\Omega _1$ is pseudoconvex in ${{\mathbb {C}}}^d \times {{\mathbb {C}}}$ and $d_{\lambda }:{{\mathbb {C}}}^d \times {{\mathbb {C}}}\rightarrow [0,\infty )$ is a distance function, we have

$$\begin{aligned} U(s,t):=-\log d\bigl ( (s,t), ({{\mathbb {C}}}^d \times {{\mathbb {C}}}) \setminus \Omega _1 \bigr ) \end{aligned}$$

is psh. Thus $U(s,0)= -\log {\widehat{\delta }}_{\lambda }(s)$ is psh. $\square$