The Lipschitz free space F(M) is a Banach space constructed “around” a given metric space M in such a way that the Lipschitz maps defined on M become linear maps on F(M). The spectacular results of Godefroy and Kalton in this direction have underlined the usefullness of this concept. At the same time the norm on F(M) is closely related to the Wasserstein distance on the probability measures on M. For these reasons the geometric properties of free spaces became a field of study in itself. In this talk we are going to survey several recent results on the interplay of the geometry of M and the geometry of F(M). For example, we will characterize the strongly exposed points of BF(M) or show that F(M) has the Daugavet property if and only if M is a length space.
Joint work with L. Garcia-Lirola, C. Petitjean and A. Rueda-Zoca.