Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded domain. Let $x_1,\ldots,x_n \in \overline{\Omega}$ be chosen so as to minimize $$ \sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|} $$ over all possible choices of $y_1,\ldots,y_n \in \overline\Omega$.

Is it then true that $\min \{|x_i - x_j| : 1\leq i < j \leq n \}$ equals the maximum of $\min\{|y_i - y_j| : 1\leq i < j \leq n\}$ over all possible choices of $y_1,\ldots,y_n \in \overline\Omega$?

I know this is related to Fekete points and the Thomson problem. Maybe it's easy. References are appreciated as well.

equilibrium measure) is known to be concentrated on $\partial \Omega$, so I suspect for large $n$ the points will tend to concentrate on the boundary. $\endgroup$ – Mateusz Kwaśnicki May 30 at 8:10