Linear extension operators between spaces of Lipschitz maps and optimal   transport
  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. 
Luigi Ambrosio, and Daniele Puglisi
math.FA