I am interested finding the collection of points in the Euclidean space that has the maximal minimal pairwise distance subject to an average norm constraint, that is, how to maximize

$min_{i \neq j} |x_i - x_j|$

subject to $\frac{1}{n} \sum_{i=1}^n |x_j|^2 \leq1$ where $\{x_1, \ldots, x_n\} \subset \mathbb{R}^d$.

I wonder if this problem has a name and what is known about it. Of course $d = 1$ is easy: just choose $n$ uniformly spaced points that satisfies the constraint with equality. I am especially interested in $d=2$. If little is known in the non-asymptotic case, maybe we know more when $n$ and/or $d$ is large? Is it related to sphere packing?

(BTW, I heard that the answer is given by vertices on the simplex when $n \leq d -1$ (or maybe the other way around?))