A weighted extremal function and equilibrium measure

Abstract

Let $K={\mathbf{R}}^n\subset {\mathbf{C}}^n$ and $Q(x):=\frac{1}{2}\log (1+x^2)$ where $x=(x_1,...,x_n)$ and $x^2=x_1^2+\cdots +x_n^2$. Utilizing extremal functions for convex bodies in ${\mathbf{R}}^n\subset {\mathbf{C}}^n$ and Sadullaev's characterization of algebraicity for complex analytic subvarieties of ${\mathbf{C}}^n$ we prove the following explicit formula for the weighted extremal function $V_{K,Q}$: $$V_{K,Q}(z)=\frac{1}{2}\log ([1+|z{|}^2]+\{[1+|z{|}^2{]}^2-|1+z^2{|}^2{\}}^{1/2})$$ where $z=(z_1,...,z_n)$ and $z^2=z_1^2+\cdots +z_n^2$. As a corollary, we find that the Alexander capacity $T_{\omega }(\mathbf{R}{\mathbf{P}}^n)$ of $\mathbf{R}{\mathbf{P}}^n$ is $1/\sqrt{2}$. We also compute the Monge-Ampère measure of $V_{K,Q}$: $$(d{d}^c{V}_{K,Q}{)}^n=n!\frac{1}{(1+x^2{)}^{\frac{n+1}{2}}}dx.$$