On the Sičiak extremal function for real compact convex sets
The Sičiak extremal function (or the pluricomplex Green function with logarithmic pole at infinity) of a compact subset \(K\) of \(\mathbb{C}^N\) is the upper semicontinuous regularization of the function \[ V_K(z)= \sup\{u(z): u\in \text{PSH}(\mathbb{C}^N), \;u(\cdot)-\log^+|\cdot|<\text{const.},\;u|_K\leq 0\}. \] It follows from results of M. Lundin and M. Baran that if \(K\) is a convex body in \(\mathbb{R}^N\), symmetric with respect to the origin, then \[ V_K(z)=\sup_l V_{l(K)}(l(z)), \tag{1} \] where \(l\) runs the set of all non-zero linear functionals on \(\mathbb{R}^N\) (extended to \(\mathbb{C}^N\) via \(l(x+iy)=l(x)+il(y)\)). The authors show that for \(z\in \mathbb{R}^N\) the relation remains true without assuming \(K\) to be symmetric (the proof is based on a result from \textit{A. Kroó} and \textit{D.~Schmidt} [J. Approximation Theory 90, 415-434 (1997; Zbl 0894.41013)]. In particular, this gives a nice description of \(V_K(x)\), \(x\in \mathbb{R}^N\), for polytops \(K\subset\mathbb{R}^N\). On the other hand, it is proved that if \(K\subset\mathbb{R}^N\) is polynomially convex and \(V_K(z)\) satisfies (1) for all \(z\in \mathbb{C}^N\) (even with \(l\) running over all complex linear functionals on \(\mathbb{C}^N\)), then \(K\) must be lineally convex. However the lineal convexity is not sufficient, which is shown by studying the case of the standard simplex in \(\mathbb{R}^2\).
pluricomplex Green function with logarithmic pole at infinity
scientific article; zbMATH DE number 1987651
scientific article