Let $(X,d)$ be a compact metric space, $m$ be a metric outer-measure on $X$. Are there 'mild conditions' on $X$ ensuring the existence of a positive integer $N\geq 3$ such that there exist $x_1,\dots,x_N\in X$ satisfying $$ m\big(\big\{ x\in X:\, \exists x_0\in X-\{x\}\; (\forall i=1,\dots,N)\quad d(x,x_i)=d(x_0,x_i) \big\}\big)=0. $$ In other words, the Kuraotwski-esque map $X\ni x\mapsto (d(x,x_n))_{n=1}^N\in \mathbb{R}^N$ is injective outside a set of metric outer-measure $0$.
Note: If $N$ were to be infinite, then the above embedding is the usual embedding of $X$ into the Hilbert cube $[0,1]^{\omega}$, which is known to be an isometry. So I would expect that, the set of 'bad points' asymptotically vanishes.
