On transfinite diameters in C^d for generalized notions of degree - Levenberg, Norman; Wielonsky, Franck (MATHEMATICA SCANDINAVICA, 2021)
We give a general formula for the C-transfinite diameter δ_C(K) of a compact set KC^2 which is a product of univariate compacta where C(R^+)^2 is a convex body. Along the way we prove a Rumely type formula relating δ_C(K) and the C-Robin function ρ_V_C,K of the C-extremal plurisubharmonic function V_C,K for C (R^+)^2 a triangle T_a,b with vertices (0,0), (b,0), (0,a). Finally, we show how the definition of δ_C(K) can be extended to include many nonconvex bodies CR^d for d-circled sets KC^d, and we prove an integral formula for δ_C(K) which we use to compute a formula for δ_C(B) where B is the Euclidean unit ball in C^2.