# Monotone Lipschitz embedding ?

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.

• Can you supply a reference? Is Aharoni's proof constructive? Can you give an example of such an embedding? – Dave Penneys Nov 29 '09 at 5:29
• The original reference is: I. Aharoni, Every separable metric space is Lipschitz equivalent to a subset of c_0, Israel J. Math. 19 (1974), 284-291. Yes, Aharoni's proof is constructive. Another reference is the Kalton & Lancien preprint from 2007 arxiv.org/abs/0708.3924 – Ady Nov 29 '09 at 6:18
• I just thought it might be useful to strip the construction to the bones (at the cost of relaxing the Lipschitz constant). Let $X$ be a separable metric space and let $x_j$ 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 $K_1(x)=\|x\|_1$ – fedja Jan 23 '10 at 0:59

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 $c_0$.