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LetM=(W, d) be a metric space. LetL 1 denote theL 1 metric. AnL 1-embedding ofM into Cartesiank-space ℝ k is a distance-preserving map from (W, d) into (ℝ k ,L 1). Letc(k) be the smallest integer such that for every metric spaceM, M isL 1-embeddable inR k iff everyc(k)-sized subspace ofM isL 1-embeddable inR k. A special case of a theorem of Menger (see p. 94 of [5]) says thatc(1) exists and equals 4. We show thatc(2) exists and satisfies 6≦c(2)≦11. Whether or notc(k) exists for anyk≧3 is an open question.

A bounded compactness theorem forL 1-embeddability of metric spaces in the plane