We give a pluripotential-theoretic proof of the product property for the transfinite diameter originally shown by Bloom and Calvi. The main tool is the Rumely formula expressing the transfinite diameter in terms of the global extremal function.
It is proved that any subharmonic function in a domain Ω ⊂ ℂⁿ which is plurisubharmonic outside of a real hypersurface of class C¹ is indeed plurisubharmonic in Ω.
We consider the following problem: find on a plurisubharmonic function with a given order function. In particular, we prove that any positive ambiguous function on which is constant outside a polar set is the order function of a plurisubharmonic function.