This could be asked in more generality, but let me stick to a concrete case.

Usually one considers a fixed domain $E \subset \mathbb{C}$ and attaches to it the equilibrium probability measure $\nu_E$, the one that minimizes the energy integral
$$
I_E(\nu) := \int_E \int_E -\log{|z-w|} \, d\nu(z) \, d\nu(w).
$$
I want to look at the opposite problem. Fix an infinite measure $\mu$ on $\mathbb{C}$, say the Lebesgue measure $dx\, dy$, and look for the measurable set $E \subset \mathbb{C}$ of unit measure $\mu(E) = 1$ for which the energy $I_E(\mu)$ is a *maximum*.

Is there a literature available on this type of question? In particular, what is the answer when $\mu$ is the Lebesgue measure? I am fine with assuming $E$ is a 'nice' domain of unit area rather than a general measurable set.

The same question could be posed in Euclidean space of any dimension, or a more general Riemannian manifold. For dimension one and the Lebesgue measure on $\mathbb{R}$ the maximum is clearly attained for a unit interval, with energy $$ \int_0^1\int_0^1 -\log{|x-y|} \, dx \, dy = 3/2. $$