TITLE: minimum space dimension to place n-points knowing pairwise distances
QUESTION [5 upvotes]: Hi everyone,
Let P be a set of n points.
Assuming I know the pairwise distances for each pair of points.
What would be the minimum dimension of the space in which I could place those n points with respect to the different pairwise distances.
The idea would be to set a first point at random coordinates in a multi-dimensional space.
Then, add the other n-1 points so that the pairwise distances are respected.
Sorry, it's maybe a trivial question for mathematicians but I'm still wondering if the  relation: "number of points-> minimum dimension of space" does exist.
Thank you for your comments.

REPLY [4 votes]: Obviously, for some special distances, you can embed the points in a fewer-dimensional space. If this is important, one way to count the correct dimension would be the rank of a certain matrix.
$(b-a,c-a)=(d(a,b)^2+d(a,c)^2-d(b,c)^2)/2$
If you fix a point $a$ and place the other values of this dot product into an $n-1$ by $n-1$ symmetric matrix, the rank of that matrix will be the dimension of the space you can embed the points in.