On the optimality of gluing over scales

We show that for every $α> 0$, there exist $n$-point metric spaces (X,d) where every "scale" admits a Euclidean embedding with distortion at most $α$, but the whole space requires distortion at least $Ω(\sqrt{α\log n})$. This shows that the scale-gluing lemma [Lee, SODA 2005] is tight, and disproves a conjecture stated there. This matching upper bound was known to be tight at both endpoints, i.e. when $α= Θ(1)$ and $α= Θ(\log n)$, but nowhere in between. More specifically, we exhibit $n$-point spaces with doubling constant $λ$ requiring Euclidean distortion $Ω(\sqrt{\log λ\log n})$, which also shows that the technique of "measured descent" [Krauthgamer, et. al., Geometric and Functional Analysis] is optimal. We extend this to obtain a similar tight result for $L_p$ spaces with $p > 1$.

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