Let $X = \{1, 2, 3, ..., k\}$ with the discrete metric (distance is 1 for every pair of points). How can this be embedded into $\mathbb{R}^n$ (with the usual metric) such that the embedding would be as close to an isometry as possible?
I have tried to formulate the question better like this:
Let $\mathcal{F} = \{ f : X \to \mathbb{R}^n \mid f \ \text{ is an embedding} \}$
and now we're asking for the function $f \in \cal{F}$ that would minimize the quantity
$ \max ( | 1 - |f(s)-f(t)| | ) $ where the maximum is over all pairs ${s,t} \in X, s\neq t.$
I think the minimizer exists, since $\cal{F}$ is basically just $\mathbb{R}^{kn} \setminus \{x \mid x \text{ has a repeated coordinate}\}$ and we can fix one point to go to the origin e.g $f(1) = 0$ and then other values of f should be about 1 unit away from this so we can probably restrict to some compact set (?).
For example: how can one place 4 points to the plane s.t. the distances between two points are all as close to 1 as possible. If you place them to the edges of a square of sidelenght $a$, the distances are $a$ (four times) and $\sqrt2 a$ (two times). We can dilate this s.t these would be equally close to 1. But is the square the best possible formation?
Alternatively we can alter the question to ask for that the distances (between two points) are all close to each other and ask to minimize the biggest over the smallest or to minimize their variance (or what ever other ways to measure there are). I don't know if these lead to the same question or not. The absolute closeness to 1 seemed most natural to me.
People usually consider bilipschitz embeddings with parameter $D$:
map $f:X \to Y$ is $D$-bilipschitz if there exists $r$ such that $r\rho_Y(f(x),f(y)) \leq \rho_X(x,y) \leq Dr\rho(f(x),f(y))$.
Now in your case finding $D$-bilipschitz embedding is finding $max(D-1, 1/D - 1)$ embedding in your case. There are a lot of results in this direction. For example Johnson-Lindenstrauss flattening lemma says that if $X \subset \mathbb{R^n}$ for some $n$ (true in your case) then for every $0<\epsilon<1$ there exists $1+\epsilon$ embedding of $X$ in $\mathbb{R}^k$ for $k=O(\epsilon^{-2}\log n)$. The proof is probabilistic and one can obtain concrete bounds.
There is a very nice result called Bourgain embedding theorem that finds a embedding in $l_1^k$ such that $||F(x)-F(y)|| \leq d(x,y) \leq ||F(x)-F(y)||c\log k$ for some absolute constant c.This can be used to obtain $l_2$ -bounds.
Hope, some of this will help to find some references or related results.