ON TRANSFINITE DIAMETERS IN Cd FOR GENERALIZED NOTIONS OF DEGREE

We give a general formula for the C-transfinite diameter delta(C) (K) of a compact set K subset of C-2 which is a product of univariate compacta where C subset of (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 of (R+)(2) a triangle T-a,T-b with vertices (0, 0), (b, 0), (0, a). Finally, we show how the definition of delta(C) (K) can be extended to include many nonconvex bodies C subset of R-d for d -circled sets K subset of C-d, and we prove an integral formula for delta(C) (K) which we use to compute a formula for delta(C) (E) where B is the Euclidean unit ball in C-2.