Pluripotential theory for convex bodies in \(\mathbb R^N\)
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.
scientific article; zbMATH DE number 2165572
scientific article