Linear extension operators between spaces of Lipschitz maps and optimal transport

Motivated by the notion ofK-gentle partition of unity introduced in [J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Invent. Math. 160 (2005), no. 1, 59-95] and the notion of K-Lipschitz retract studied in [S. I. Ohta, Extending Lipschitz and Hölder maps between metric spaces, Positivity 13 (2009), no. 2, 407-425], 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.