The Johnson Lindenstrauss Lemma (pasted below from a source) seems to me to have a strange implication.
Suppose I have two points (i.e. $n=2$) and I consider a very coarse error tolerance (e.g. $\varepsilon = 0.49)$. I think the missing constant obscured by $O()$ is 20, but this makes the following implication worse. The lemma says that there exists a linear map requiring $m = 0.49^{-2} \log(2) \approx 2.88 $ dimensions to preserve the distance between these two points. How does this make sense? Surely one dimension would suffice, no?
The lemma says that such a map exists for the given value of $m$. It does not say that you can't also get a map for a smaller value of $m$. In other words, it makes no claim that the given value of $m$ is optimal.