d Abstract) Sanjeev Arora∗ L´aszl´o Lov´asz† Ilan Newman‡ Yuval Rabani§ Yuri Rabinovich¶ Santosh Vempala Motivated by applications in combinatorial optimization, we initiate a study of the extent to which the global properties of a metric space (especially, embeddability in 1 with low distortion) are determined by properties of “small” sets. We prove both positive and negative results.