2 dimensional representations of v dimensional points based on distance

Question

I have a set of points $x_1, \dots, x_n \in \mathbb R^v$

I have a measure of the distance between each one of these points $D \in \mathbb R^{n\times n}$ where $D_{i,j}= distance(x_i, x_j)$

I would like to visualize these points in a plan in such a way that the euclidean distance in the plan is equal or proportional to the distance stored in the matrix $D$ .

Is there any way to do that? How?

Answer 1

This is not always possible, think of the four vertices $x_i$, $i = 1, \ldots, 4$ of a regular tetrahedron $\subseteq \mathbf R^3$. Then we have $\def\abs#1{\left|#1\right|}\abs{x_i - x_j} = 1$ for $i \ne j$, but there don't exist four points in the plane having mutual distance 1.