The extremal psh for the complement of convex, symmetric subsets of \({\mathbb{R}}^ N\) (Q1065209)

For a compact subset E of \({\mathbb{C}}^ n\) the extremal function is defined by \(\phi_ E(z)=\sup \{| p(z)| \}^{1/\deg p},\) the supremum being taken over all polynomials in n complex variables with \(\| p\|_ E\leq 1\). This function has been investigated in connection with polynomial approximation by Zahariuta and Siciak, and in connection with the complex Monge-Ampere equation by Bedford and Taylor. In this note a fairly explicit formula for \(\phi_ E\) is given when E is convex, symmetric with respect to 0 and contained in \({\mathbb{R}}^ n\subset {\mathbb{C}}^ n\). The calculation gives also a complex foliation of \({\mathbb{C}}^ n\) such that \(\phi_ E\) is harmonic on each leaf except at the intersection between the leaves and E.