Gamow liquid-drop minimizer conjecture

For a measurable set $\Omega\subset\mathbb R^3$, let $\mathcal E(\Omega)=P(\Omega)+\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|},$ where $P$ is De Giorgi perimeter, and set $V_*=5\frac{2-2^{2/3}}{2^{2/3}-1}\approx3.51.$ The conjecture asks for the complete fixed-volume minimization picture. Chodosh and Gianocca prove that, for every $0<V\le V_*$, balls of volume $V$ uniquely minimize $\mathcal E$ among all measurable $\Omega$ with $|\Omega|=V$, up to translation and null sets; for $V>V_*$, no minimizer exists. Consequently, $\inf_{0<|\Omega|<\infty}\frac{\mathcal E(\Omega)}{|\Omega|}=3\left(\frac{9\pi}{5}\right)^{1/3}=\frac92\left(\frac{8\pi}{15}\right)^{1/3},$ with equality exactly for translates, modulo null sets, of the ball of volume $5/2$, equivalently radius $(15/(8\pi))^{1/3}$.