关于标度上胶合的最优性
On the optimality of gluing over scales
摘要：
我们证明了对于每个$α> 0$，存在$n$ -点度量空间（X，d），其中每个“尺度”允许最多$α$的欧氏嵌入，但整个空间要求至少$Ω(\sqrt{α\log n})$的失真.这表明标度粘合引理[Lee，SODA 2005]是紧的，并反驳了那里的一个猜想。已知该匹配上限在两个端点处都是紧的，即当$α= Θ(1)$和$α= Θ(\log n)$时，但在两者之间没有任何地方。更具体地说，我们展示了$n$ -点空间与加倍常数$λ$需要欧几里德失真$Ω(\sqrt{\log λ\log n})$，这也表明，技术的“测量下降”[Krauthgamer等。例如，几何和功能分析]是最佳的。我们推广这一结果，得到了类似的紧的结果与$p > 1$的$L_p$空间。
英文摘要：
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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