  Motivated by the notion of K-gentle partition of unity introduced in [12] and
the notion of K-Lipschitz retract studied in [17], we study a weaker notion
related to the Kantorovich-Rubinstein transport distance, that we call K-random
projection. We show that K-random projections can still be used to provide
linear extension operators for Lipschitz maps. We also prove that the existence
of these random projections is necessary and sufficient for the existence of
weak* continuous operators. Finally we use this notion to characterize the
metric spaces (X, d) such that the free space F(X) has the bounded
approximation propriety.
