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

Geometry and volume product of finite dimensional Lipschitz-free spaces