A functional representation of almost isometries

For each quasi-metric space X   we consider the convex lattice SLip1(X)SLip1(X) of all semi-Lipschitz functions on X with semi-Lipschitz constant not greater than 1. If X and Y are two complete quasi-metric spaces, we prove that every convex lattice isomorphism T   from SLip1(Y)SLip1(Y) onto SLip1(X)SLip1(X) can be written in the form Tf=c⋅(f∘τ)+ϕTf=c⋅(f∘τ)+ϕ, where τ   is an isometry, c>0c>0 and ϕ∈SLip1(X)ϕ∈SLip1(X). As a consequence, we obtain that two complete quasi-metric spaces are almost isometric if, and only if, there exists an almost-unital convex lattice isomorphism between SLip1(X)SLip1(X) and SLip1(Y)SLip1(Y).