The critical-volume conjecture for the nonlocal isoperimetric problem in Euclidean space

Let $n\geq 2$, $0<\alpha<n$, and $\gamma>0$. For a measurable set $F\subset\mathbb{R}^n$ with $|F|=m$, let $\mathcal{E}(F)$ denote the nonlocal isoperimetric functional, and let $m_{*}$ be the critical volume defined by equality between the energy of a ball of volume $m_{*}$ and that of two infinitely separated balls, each of volume $m_{*}/2$.

Critical-volume conjecture. For $m\leq m_{*}$, the ball of volume $m$ uniquely minimizes $\mathcal{E}(\cdot)$ among measurable sets $F\subset\mathbb{R}^n$ with $|F|=m$, and for $m>m_{*}$ no minimizer exists.

The conjecture predicts a sharp transition at the critical volume: below it, the perimeter-dominated problem has a unique spherical minimizer, while above it, splitting into widely separated components prevents attainment. Its status is not resolved in the supplied source context.