  Andoni, Krauthgamer and Razenshteyn (AKR) proved (STOC 2015) that a
finite-dimensional normed space $(X,\|\cdot\|_X)$ admits a $O(1)$ sketching
algorithm (namely, with $O(1)$ sketch size and $O(1)$ approximation) if and
only if for every $\varepsilon\in (0,1)$ there exist $\alpha\geqslant 1$ and an
embedding $f:X\to \ell_{1-\varepsilon}$ such that $\|x-y\|_X\leqslant
\|f(x)-f(y)\|_{1-\varepsilon}\leqslant \alpha \|x-y\|_X$ for all $x,y\in X$.
The "if part" of this theorem follows from a sketching algorithm of Indyk (FOCS
2000). The contribution of AKR is therefore to demonstrate that the mere
availability of a sketching algorithm implies the existence of the
aforementioned geometric realization. Indyk's algorithm shows that the "if
part" of the AKR characterization holds true for any metric space whatsoever,
i.e., the existence of an embedding as above implies sketchability even when
$X$ is not a normed space. Due to this, a natural question that AKR posed was
whether the assumption that the underlying space is a normed space is needed
for their characterization of sketchability. We resolve this question by
proving that for arbitrarily large $n\in \mathbb{N}$ there is an $n$-point
metric space $(M(n),d_{M(n)})$ which is $O(1)$-sketchable yet for every
$\varepsilon\in (0,\frac12)$, if $\alpha(n)\geqslant 1$ and $f_n:M(n)\to
\ell_{1-\varepsilon}$ are such that $d_{M(n)}(x,y)\leqslant
\|f_n(x)-f_n(y)\|_{1-\varepsilon}\leqslant \alpha(n) d_{M(n)}(x,y)$ for all
$x,y\in M(n)$, then necessarily $\lim_{n\to \infty} \alpha(n)= \infty$.
