Balls are maximizers of the Riesz-type functionals with supermodular integrands (Q1956514)

The Riesz-type functional \[ J(f,g) := \int_{\mathbb R^n\times\mathbb R^n} \Psi\big(f(x), g(x)\big) h\big(|x-y|\big) dx\,dy \] is considered. It is shown that for \(\Psi\) being a supermodular operator satisfying some extra assumptions, the euclidean balls are the unique maximizers of \(J(f,g)\).