Open problems on embeddings of finite metric spaces

2 Problems from 2002 4 2.1 Subspaces of l1 into l n 1 (Gideon Schechtman) . . . . . . . . . . . 4 2.2 Squared l2 metrics into l1 (Nathan Linial) . . . . . . . . . . . . . 5 2.3 Girth and l1/l2 embeddings (Nathan Linial) . . . . . . . . . . . 5 2.4 Algorithmic difficulty of l1-embeddings (Chandra Chekuri; Anupam Gupta) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Planar-graph metrics into l1 (Nathan Linial) . . . . . . . . . . . 6 2.6 Decision problem (Yuri Rabinovich) . . . . . . . . . . . . . . . . 7 2.7 How large graph? (Jǐŕı Matoušek; probably folklore) . . . . . . . 7 2.8 Planar into R2 (Jǐŕı Matoušek) . . . . . . . . . . . . . . . . . . . 7 2.9 Algorithmic complexity (Chandra Chekuri) . . . . . . . . . . . . 7 2.10 Bandwidth and ball growth (Nathan Linial) . . . . . . . . . . . . 8 2.11 Into R3 (Jǐŕı Matoušek) . . . . . . . . . . . . . . . . . . . . . . . 8 2.12 Lipschitz mapping of n grid points onto a square (Uriel Feige) . . 8 2.13 Path into R3, volume-respecting (Uriel Feige) . . . . . . . . . . . 9 2.14 Path into l2 (Nathan Linial) . . . . . . . . . . . . . . . . . . . . . 9 2.15 Levenstein metric into l1 (Piotr Indyk) . . . . . . . . . . . . . . . 9 2.16 Fréchet metric into l∞ (Piotr Indyk) . . . . . . . . . . . . . . . . 10 2.17 Earth-mover distance (Piotr Indyk) . . . . . . . . . . . . . . . . . 11 2.18 Convex extensions (Yuval Rabani) . . . . . . . . . . . . . . . . . 11 2.19 Explicit graphs with high sphericity (Nathan Linial) . . . . . . . 12

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