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.

中文翻译：

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有限维 Lipschitz 自由空间的几何和体积积

本文的目标是研究凸体 $B_{\mathcal F(M)}$ 的几何和极值性质，它是与有限度量空间 $M$ 相关的 Lipschitz-free Banach 空间的单位球。我们研究了 $\ell_1$ 和 $\ell_\infty$-sums，特别是我们刻画了度量空间，使得 $B_{\mathcal F(M)}$ 是 Hanner 多胞体。我们还刻画了有限度量空间，其 Lipschitz 自由空间是等距的。我们讨论了体积积 $\mathcal{P}(M)=|B_{\mathcal F(M)}|\cdot|B_{\mathcal F(M)}^\circ|$ 的极端性质，当$M$ 的元素数量是固定的。我们证明如果 $\mathcal P(M)$ 在具有相同点数的所有度量空间中最大，那么 $M$ 中的所有三角形不等式都是严格的，并且 $B_{\mathcal F(M)}$ 是单纯的.