We show that, for a separable and complete metric space M , the Lipschitz-free space F(M) embeds linearly and almost-isometrically into 1 if and only if M is a subset of an R-tree with length measure 0. Moreover, it embeds isometrically if and only if the length measure of the closure of the set of branching points of M (taken in any minimal R-tree that contains M) is negligible. We also prove that, for any subset M of an R-tree, every extreme point of the unit ball of F(M) is an element of the form (δ(x) − δ(y))/d(x, y) for x = y ∈ M .

中文翻译：

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将 Lipschitz-free 空间嵌入ℓ1

我们证明，对于可分离且完备的度量空间 M ，当且仅当 M 是长度测度为 0 的 R 树的子集时，Lipschitz 自由空间 F(M) 线性地且几乎等距地嵌入到 1 中。此外，当且仅当 M 的分支点集合的闭包的长度度量（在包含 M 的任何最小 R 树中获取）可忽略不计时，它才会等距嵌入。我们还证明，对于 R 树的任何子集 M，F(M) 的单位球的每个极值点都是形式为 (δ(x) − δ(y))/d(x, y ) 对于 x = y ∈ M 。