## Minimizing variance of distances between points when mean distance is fixed

In Rd, I have n > d+1 points. The mean distance between pairs of points is 1. How can I minimize the variance of the distances (equivalently, the mean squared distance)? I'm mainly interested in d ∈ {1,2,3} although I'd be curious if there were any patterns for larger values of d. As for values of n, I'm interested both in specific solutions for small values, and general patterns in larger values.

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Pairs of distinct points, or all pairs of point? If the latter, do you count (y,x) separately from (x,y)? – Ricky Demer Aug 8 2010 at 6:51
Sorry, good point (no pun intended). I meant pairs of distinct points. – Robin Saunders Aug 8 2010 at 11:43
It's of small consequence how you count the points since, for fixed n, it will only scale the mean 1 by a constant, and this will not change the variance-minimizing configuration. What might be significant is whether you intend the arithmetic, geometric, harmonic, ... mean (only joking!). – John Bentin Aug 8 2010 at 12:18

For $d=1$, it is minimized by arithmetic progression.