Choksi–Peletier conjecture on balls and dichotomy for the liquid-drop problem

For $n=3$ and $\alpha=2$, consider the liquid-drop energy $P(E)+\mathcal{V}_\alpha(E)$ among sets $E\subset\mathbb{R}^n$ of prescribed volume. A minimizer is a set attaining the infimum of this energy. Choksi–Peletier conjecture. Whenever a minimizer exists, it is a ball; when no minimizer exists, the infimum is attained, in the sense of configurations escaping to infinity, by finitely many balls of equal volume that are infinitely far apart. The conjecture describes the expected ball-versus-dichotomy behavior of liquid-drop minimizers. In any dimension, the claim is known for $\alpha$ sufficiently close to $n$ in the result cited by the authors; the paper reports numerical evidence for the full range in dimension two.