More precisely I would like to consider the following problem:


Let $\{a_i\}_{i=1}^n$ be n points in $\mathbb{R}^3$ and assume I have the constraint $\min_{i \neq j} |a_i-a_j| = r_{min}$. How can I place $n$ points in $\mathbb{R}^3$ as to minimize $\sum_{i \neq j} |a_i-a_j|.$

When $n=2$ this is *trivial*. When $n=3$ the minimizer will clearly be an *equilateral triangle* with a point at each vertex. Already when $n=4$ the answer is *not clear* to me.

I presume that if this is known it is a well established result in graph theory and I would appreciate any references.