Given a finite metric space M, the set of Lipschitz functions on M with Lipschitz constant at most 1 can be identified with a convex polytope, say K(M), in R^n. In this talk, we will show that there is a strong connection between the geometric properties of this polytope (as the vertices or the volume product) and the properties of the metric space M. We will also relate this study with a famous open problem in Convex Geometry, the Mahler conjecture, on the product of the volume of a convex body and its polar. This is a joint work with M. Alexander, M. Fradelizi, and A. Zvavitch.