$C-$Robin Functions and Applications

We continue the study in the setting of pluripotential theory arising from polynomials associated to a convex body $C$ in $({\mathbf{R}}^{+}{)}^d$. Here we discuss $C-$Robin functions and their applications. In the particular case where $C$ is a simplex in $({\mathbf{R}}^{+}{)}^2$ with vertices $(0,0),(b,0),(a,0)$, $a,b>0$, we generalize results of T. Bloom to construct families of polynomials which recover the $C-$extremal function $V_{C,K}$ of a nonpluripolar compact set $K\subset {\mathbf{C}}^d$.