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Andoni, Krauthgamer and Razenshteyn (AKR) proved (STOC 2015) that afinite-dimensional normed space $(X,\|\cdot\|_X)$ admits a $O(1)$ sketchingalgorithm (namely, with $O(1)$ sketch size and $O(1)$ approximation) if andonly if for every $\varepsilon\in (0,1)$ there exist $\alpha\geqslant 1$ and anembedding $f:X\to \ell_{1-\varepsilon}$ 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 (FOCS2000). The contribution of AKR is therefore to demonstrate that the mereavailability of a sketching algorithm implies the existence of theaforementioned geometric realization. Indyk's algorithm shows that the "ifpart" 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 waswhether the assumption that the underlying space is a normed space is neededfor their characterization of sketchability. We resolve this question byproving that for arbitrarily large $n\in \mathbb{N}$ there is an $n$-pointmetric 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 $\lim_{n\to \infty} \alpha(n)= \infty$.

The Andoni--Krauthgamer--Razenshteyn characterization of sketchable norms fails for sketchable metrics