C-transfinite diameter

We give a general formula for the $C-$transfinite diameter ${\delta }_C(K)$ of a compact set $K\subset {\mathbb{C}}^2$ which is a product of univariate compacta where $C\subset ({\mathbb{R}}^{+}{)}^2$ is a convex body. Along the way we prove a Rumely type formula relating ${\delta }_C(K)$ and the $C-$Robin function ${\rho }_{V_{C,K}}$ of the $C-$extremal plurisubharmonic function $V_{C,K}$ for $C \subset ($\mathbb{R}$^+)^2$atriangle$T_{a,b}$withvertices$(0,0), (b,0), (0,a)$\mathrm{.}Finally,weshowhowthedefinitionof$\delta_C(K)$ can be extended to include many nonconvex bodies $C\subset {\mathbb{R}}^d$ for $d-$circled sets $K\subset {\mathbb{C}}^d$, and we prove an integral formula for ${\delta }_C(K)$ which we use to compute a formula for the $C-$transfinite diameter of the Euclidean unit ball $\mathbb{B}\subset {\mathbb{C}}^2$.