Pluripotential theory for convex bodies in \(\mathbb R^N\)
type: Item
instance of: scholarly article
title: Pluripotential theory for convex bodies in \(\mathbb R^N\)
zbMATH Open document ID: 1071.32025
published in: Mathematische Zeitschrift
publication date: 2005-05-04T00:00:00Z
review text: Let \(K\subset{\mathbb C}^N\) be a compact, convex body, and let \(V_K(z)\) be the Siciak-Zaharjuta extremal function. Then, as shown by Lundin and later Baran, if \(K\) is symmetric with respect to the origin, i.e., \(K=-K\), then the complement of \(K\) in \({\mathbb C}^N\) is foliated, in a continuous manner, by one-dimensional analytic disks \(L\) (leaves) such that \(V_K\) restricted to each leaf is harmonic. The main goal of this note is to show that a version of Lundin's result remains valid without the symmetric hypothesis. Indeed, as in the symmetric case, the leaves \(L\) are complex ellipses. The main idea is to approximate \(K\) by a decreasing sequence of compact sets \(\{K_j\}\) in \({\mathbb C}^N\) of the form \(K_j=\overline{D}_j\), where each \(D_j\) is a strictly convex domain. For each \(D_j\) one can apply the Lempert theory to get analytic disks \(L_j\) such that \(V_{K_j}\) restricted to each \(L_j\) is harmonic. Taking limits as \(j\to\infty\) the authors obtain this result.
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32U15
Mathematics Subject Classification ID: 31C10
zbMATH DE Number: 2165572
zbMATH Keywords: Siciak-Zaharjuta extremal function
zbMATH Keywords: plurisubharmonic function
zbMATH Keywords: convex body
zbMATH Keywords: analytic disk
zbMATH Keywords: complex ellipses
zbMATH Keywords: Lempert theory
author: Daniel Burns
author: Norm Levenberg
author: Sione Ma'u
MaRDI profile type: Publication
full work available at URL: s00209-004-0743-z
OpenAlex ID: W2101371192
cites work: Q4276253
cites work: Siciak's extremal function of convex sets in $C^N$
cites work: Plurisubharmonic extremal functions and complex foliations for the complement of convex sets in \(\mathbf R^ n\)
cites work: Foliations and complex monge-ampère equations
cites work: Robin functions and extremal functions
cites work: On the Sičiak extremal function for real compact convex sets
cites work: Q4002495
cites work: Metrics Associated with Extremal Plurisubharmonic Functions
cites work: La métrique de Kobayashi et la représentation des domaines sur la boule
cites work: Symmetries and other transformations of the complex Monge-Ampère equation
cites work: The extremal psh for the complement of convex, symmetric subsets of \({\mathbb{R}}^ N\)
cites work: Plurisubharmonic functions characterized by one-variable extremal functions
cites work: An extremal plurisubharmonic function associated to a convex pluricomplex Green function with pole at infinity.
cites work: Q4735594
cites work: Q4328333
cites work: Extremal plurisubharmonic functions in $C^N$
cites work: Maximal convergence of sequences of harmonic polynomials
DOI: 10.1007/S00209-004-0743-Z
description: scientific article; zbMATH DE number 2165572