minimum space dimension to place n-points knowing pairwise distances 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.
 A: Suppose your $\binom{n}{2}$ distances are all exactly 1.  Then they determine an $(n{-}1)$-simplex
in $\mathbb{R}^{n-1}$, and those distances cannot be realized in a lower dimension.
For example, three points at unit distance determine an equilateral triangle in $\mathbb{R}^{2}$;
four points at unit distance determine a regular tetrahedron in $\mathbb{R}^{3}$, whose six edge
lengths are each 1.
            

Generally, the more interesting question is how to embed the distances in a space
of modest dimension without significant distortion.
See, e.g., the chapter by Indyk and Matoušek,
"Low-distortion embeddings of finite metric spaces,"
Handbook of Discrete and Computational Geometry, 2004, or
the 2006 paper by Bartal, "Embedding finite metric spaces in low dimension."
A: 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.
A: For the basic result, start with 
Wikipedia
or Google "Johnson-Lindenstrauss Lemma".
