Given $N$ equally charged points lying on the unit sphere ("electrons"), the Thomson problem consists of finding the configuration of these points such that the electrostatic potential energy
$$ U=k_e\sum_{i<j}\frac{q^2}{|r_i-r_j|} \bigg(\text{normalize to } U=\sum_{i<j}\frac{1}{|r_i-r_j|}\bigg) $$
is minimized.
Finding such a configuration is an open problem except for some few $N$.
My question is what if one maximizes
$$ U=\sum_{i<j}|r_i-r_j| ? $$
Q1: Is this a physical concept like the electrostatic potential energy in the other case?
Q2: Is this problem equivalent to the one with $\frac{1}{|r_i-r_j|}$? Is it an open problem?
Q3: In this question there is a formula for the optimal value of $\sum_{i<j}|r_i-r_j|$, namely $$ U=\sum_{i<j}|r_i-r_j| = \frac{2}{3}N^2R $$ but I find that this is wrong: Take N=3, then the optimized value should be 6. However, inside a sphere of radius 1, the maximum distance between two points is 2, so the distances must be $d_{1,2}=d_{1,3}=d_{2,3}=2$. This is impossible because $d_{1,2}=d_{2,3}=2$ forces 1 and 3 to be at the opposite point of 2, but then 1 and 3 are at the same point, so $d_{1,3}=0$
Is my counterexample wrong, or is the formula wrong?
