Pluripotential Energy

For probability measures $μ$ on compact subsets of $\CC^n$ we define two functionals $J(μ)$ and $W(μ)$ modeled on discrete approximations to $μ$ and multivariate Vandermonde determinants. We show that these functionals coincide, up to a constant, with the electrostatic energy of $μ$ defined in a more general setting by Berman, Boucksom, Guedj and Zeriahi. This generalizes the classical notion of logarithmic energy of a measure in the complex plane; i.e., the case $n=1$.

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