Two-dimensional ball-cluster conjecture for the liquid-drop problem

Let $n=2$, let $\alpha\in(0,2)$, and define $f(m)=P(B[m])+\mathcal{V}_\alpha(B[m])$, where $B[m]$ is a ball of volume $m$. Set $m_0=0$, and for $k\geq1$ let $m_k$ be the critical masses defined in the paper by the equality of the energies of $k$ and $k+1$ equal-volume balls. Two-dimensional ball-cluster conjecture. If $m\in[m_{k-1},m_k]$, then

$$\inf_{E\subset\mathbb{R}^n,\,|E|=m} P(E)+\mathcal{V}_\alpha(E)=k f\left(\frac{m}{k}\right).$$

In particular, whenever a minimizer exists it is a ball, while when no minimizer exists the optimal configuration is, in an appropriate sense, $k$ equal-volume balls infinitely far apart. This is presented as an equivalent formulation of the Choksi–Peletier conjecture in dimension two; the paper gives numerical evidence for all $\alpha\in(0,2)$, while noting partial theoretical results for $\alpha$ close to $n$.