  Let $\varepsilon\in(0,1)$ and $X\subset\mathbb R^d$ be arbitrary with $|X|$
having size $n>1$. The Johnson-Lindenstrauss lemma states there exists
$f:X\rightarrow\mathbb R^m$ with $m = O(\varepsilon^{-2}\log n)$ such that $$
\forall x\in X\ \forall y\in X, \|x-y\|_2 \le \|f(x)-f(y)\|_2 \le
(1+\varepsilon)\|x-y\|_2 . $$ We show that a strictly stronger version of this
statement holds, answering one of the main open questions of [MMMR18]:
"$\forall y\in X$" in the above statement may be replaced with "$\forall
y\in\mathbb R^d$", so that $f$ not only preserves distances within $X$, but
also distances to $X$ from the rest of space. Previously this stronger version
was only known with the worse bound $m = O(\varepsilon^{-4}\log n)$. Our proof
is via a tighter analysis of (a specific instantiation of) the embedding recipe
of [MMMR18].
