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.

`$x_j$`

be a countable dense set. Let`$d_j(x)$`

be the distance from $x$ to`$x_j$`

. Put`$(Kx)_n=\min(4d_1(x),4d_2(x),\dots,4d_{n-1}(x),d_n(x))$`

. Then $\frac 13d(x,y)\le \|Kx-Ky\|\le 4d(x,y)$. Unfortunately, this is highly non-monotone. What I wonder though is if there is any monotone Lipshitz (not necessarily bi-Lipschitz) mapping $K$ with`$K_1(x)=\|x\|_1$`

$\endgroup$ – fedja Jan 23 '10 at 0:59