A weighted extremal function and equilibrium measure

Let $K={\mathbf{R}}^n\subset {\mathbf{C}}^n$ and $Q(x):=\frac{1}{2}\log (1+x^2)$ where $x=(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n)$ 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 ${\bf C}^n$ we prove the following explicit formula for the weighted extremal function $V_{K,Q}$: $$V_{K,Q}(z)=\frac{1}{2}\log ([1+\mathrm{\mid }z{\mathrm{\mid }}^2]+\{[1+\mathrm{\mid }z{\mathrm{\mid }}^2{]}^2-\mathrm{\mid }1+z^2{\mathrm{\mid }}^2{\}}^{1\mathrm{/}2})$$ where $z=(z_1,\mathrm{.}\mathrm{.}\mathrm{.},z_n)$ and $z^2 = z_1^2+\cdots +z_n^2$\mathrm{.}Asacorollary,wefindthattheAlexandercapacity$T_{\omega}({\bf R} {\bf P}^n)$of${\bf R} {\bf P}^n$is$1/\sqrt 2$. We also compute the Monge-Amp`ere measure of $V_{K,Q}$: $$(d{d}^c{V}_{K,Q}{)}^n=n!\frac{1}{(1+x^2{)}^{\frac{n+1}{2}}}dx\mathrm{.}$$