TITLE: Monotone Lipschitz embedding ?
QUESTION [11 upvotes]: 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.

REPLY [4 votes]: To answer Bill Johnson's question, a monotone linear bi-Lipschitz embedding (actually, an isometric one) $\ell^1\to\ell^\infty$ is very easy to construct. Just take any antisymmetric matrix $A$ of $\pm 1$s with the property that for each $n$ every combination of signs in the first $n$ positions appears in some row of $A$ (you can easily build it by induction) and take $Lx=x+Ax$. Unfortunately, I do not see how to convert it into a mapping to $c_0$.