Consider the following standard formulation of the Johnson-Lindenstrauss lemma: 

>Lemma (JL).<br>
>For any $0<\epsilon < 1$ and any integer $n$, let $k$ be a positive integer such that $k\geq C\cdot \epsilon^{-2}\log n$ (for some explicit constant $C$ which we omit here). Then, for any set $X$ of $n$ points in $\mathbb{R}^d$, there is a linear map $f:\mathbb{R}^d\rightarrow \mathbb{R}^k$ such that ***for all*** $u,v\in X$, it holds; 
>
>$||u-v||^2(1-\epsilon) \leq ||f(u)- f(v)||^2 \leq ||u-v||^2(1+\epsilon)$.
>
>Moreover, $f$ can be found in randomized polynomial time.

Now, what can be said about probabilistic $(1+\epsilon)$ distortion of arbitrary vectors in $\mathbb{R}^d$, that are not necessarily drawn from $X$ ? 

In particular, suppose $X$ is fixed and $f$ is computed as in JL, then we pick an arbitrary (random) $z\in \mathbb{R}^d$, not necessarily in $X$, and we look at the distortion between $z$ and $x$, for every $x\in X$: can we appropriately bound the probability to get $(1+\epsilon)$ distortion w.r.t. $z,x$ ?

I guess it is possible to appropriately bound, and then amplify, the probability $\delta$ that an arbitrary (random) $z\in \mathbb{R}^d$ will be distorted by at most $(1+\epsilon)$ relative error w.r.t $X$, as above. Is this guess true ?