Given pairwise distances of $N$ data points and find the minimal dimenion of space can fit the data Given a set $D$ consist of all pairwise distance of $N$ unknown dimension points.
e.g. If there is 3 points, ${x_a,x_b,x_c}$
$$D=\{||x_a-x_b||,||x_b-x_c||,||x_a-x_c||\}$$
How can I find the minimum dimension of space which can place all these $N$ points on it and satisfy the distance constraints?

For this topic I can recommend the pretty old paper "Discussion of a set of point in terms of their mutual distances" by Gale Young and A.S. Householder. Although it is from 1938 and uses not "up-to-date" mathematical language it is pretty accessible. By the way: The answer lies in computing the rank of one specific matrix and you can calculate a constellation of points which produce the desired distances by one SVD. 
