  The goal of this paper is to study geometric and extremal properties of the
convex body $B_{\mathcal F(M)}$, which is the unit ball of the Lipschitz-free
Banach space associated with a finite metric space $M$. We investigate $\ell_1$
and $\ell_\infty$-sums, in particular we characterize the metric spaces such
that $B_{\mathcal F(M)}$ is a Hanner polytope. We also characterize the finite
metric spaces whose Lipschitz-free spaces are isometric. We discuss the extreme
properties of the volume product
  $\mathcal{P}(M)=|B_{\mathcal F(M)}|\cdot|B_{\mathcal F(M)}^\circ|$, when the
number of elements of $M$ is fixed. We show that if $\mathcal P(M)$ is maximal
among all the metric spaces with the same number of points, then all triangle
inequalities in $M$ are strict and $B_{\mathcal F(M)}$ is simplicial. We also
focus on the metric spaces minimizing $\mathcal P(M)$, and in the Mahler's
conjecture for this class of convex bodies.
