Lipschitz quotients from metric trees and from Banach spaces containing l1

A Lipschitz map f between the metric spaces X and Y is called a Lipschitz quotient map if there is a C > 0 (the smallest such C, the co-Lipschitz constant, is denoted coLip(f), while Lip(f) denotes the Lipschitz constant of f) so that for every x ∈ X and r > 0, fBX(x, r) ⊃ BY (f(x), r/C). Thus Lipschitz quotient maps are surjective maps which by definition have the property ensured by the open mapping theorem for surjective linear operators between Banach spaces. If there is a Lipschitz quotient mapping f from X onto Y with Lip(f) · coLip(f) ≤ C, we say that Y is a C-Lipschitz quotient of X. If Y is a C-Lipschitz quotient of X for some C, we say that Y is a Lipschitz quotient of X. Lipschitz quotient maps were introduced and studied in [4]. Among other results the following proposition was proved in [4] (see also [5], Theorem 11.18).