Given an $n$-dimensional vector space, I want to find a set of $m$ vectors $(V_i)_{i = 1 .. m}$ such that the pairwise euclidean distances between each vector in the set have minimal sample variance. (Measures of variation other than the sample variance would also be fine. I am just looking for distances that are "as similar as possible" to each other.)
For a given size $m$, how can I find such a set?
Here's my thoughts so far:
- For what concerns the edge case of a set with variance zero in the distribution of pairwise distances (i.e. all vectors in the set have the same pairwise distance), I am fairly convinced that such a set is findable only if $m \le n+1$. For example: in 1 dimension, a set of points with equal pairwise distances can have at most 2 points; in 2 dimensions, a set of points with equal pairwise distances can have at most 3 points (a triangle).
- However, if I want a set with $m > n+1$ elements, I don't really know what's a good intuition to tackle this problem.