Stefan asks:
<<
Suppose we have the m different vectors x_1, x_2,
..., x_m in the R^n. No two vectors are the same and
no vector is the zero vector. Furthermore we calculate
for every 1<=i<j<=m the normal dot product x_i.x_j
How many different numbers are we at least going
to get, regardless of the choice of x_1, x_2, ..., x_m?
. . .
[W]hat if m>n? Is anything known about this problem?
Any ideas, results, bounds?
>>
Also of course if m = n+1, the vectors can be the n+1 vertices of a regular n-simplex centered at the origin, having only 1 value of
x_i*x_j for any i <>j.
I find the problem more natural if the vectors are assumed to be of constant length. Then the question is essentially:
"Given m distinct points on the surface of the unit sphere in R^n,
what is the smallest size that the set of their mutual distances can be?"
A natural generalization of this version would be to consider a set of
m points that all lie in any other highly symmetric space S -- such as
S = the n-torus (or S = any homogeneous space), and again ask how small the set of distinct distances can be.
Very nice geometric questions here.
It may be possible -- and fun -- to find good answers numerically by taking any starting configuration of the m points on the metric space S, and sum over the (m(m-1)/2)(m(m-1)/2 - 1)/2 pairs of distinct ordered pairs (i,j),(i',j') for 1 <= i < j <= m and 1 <= i' < j' <= m. Then minimize this sum via some robust minimization algorithm, like conjugate gradient, using many starting configurations.)
The spheres S^2 and S^3, the projective spaces P^2 and P^3, and the tori T^2 and T^3 would be especially interesting to work on, as well as complex projective spaces like CP^2 and compact connected Lie groups like SO(4).
--Dan
