In 1974, Aharoni proved that every separable metric space (X, d) is Lipschitz isomorphic to a subset of the Banach space c_0. Thus, for some constant L, there is a map K: X --> c_0 that satisfies the inequality d(u,v) <= || Ku - Kv || <= Ld(u,v) for all u and v in X. Now, suppose X = l_1 (in this case, L = 2 is best possible). I have the following
Conjecture: Let K: l_1 --> c_0 be a Lipschitz embedding. Then K cannot be monotone w.r.t. the natural duality pairing (.,.) between l_1 and c_0, i.e., there are some u and v in l_1 such that (u - v, Ku - Kv) < 0.