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Examples and applications of the density of strongly norm attaining Lipschitz maps
Revista Matemática Iberoamericana ( IF 1.2 ) Pub Date : 2021-02-01 , DOI: 10.4171/rmi/1253
Rafael Chiclana 1 , Luis García-Lirola 2 , Miguel Martín 1 , Abraham Rueda Zoca 3
Affiliation  

We study the density of the set SNA$(M,Y)$ of those Lipschitz maps from a (complete pointed) metric space $M$ to a Banach space $Y$ which strongly attain their norm (i.e., the supremum defining the Lipschitz norm is actually a maximum). We present new and somehow counterintuitive examples, and we give some applications.

First, we show that SNA$(\mathbb T,Y)$ is not dense in Lip$_0(\mathbb T,Y)$ for any Banach space $Y$, where $\mathbb T$ denotes the unit circle in the Euclidean plane. This provides the first example of a Gromov concave metric space (i.e., every molecule is a strongly exposed point of the unit ball of the Lipschitz-free space) for which the density does not hold.

Next, we construct metric spaces $M$ satisfying that SNA$(M,Y)$ is dense in Lip$_0(M,Y)$ regardless $Y$ but which contain isometric copies of $[0,1]$ and so the Lipschitz-free space $\mathcal F(M)$ fails the Radon–Nikodym property, answering in the negative a question posed by Cascales et al. in 2019 and by Godefroy in 2015. Furthermore, an example of such $M$ can be produced failing all the previously known sufficient conditions for the density of strongly norm attaining Lipschitz maps.

Finally, among other applications, we show that if $M$ is a boundedly compact metric space for which SNA$(M,\mathbb R)$ is dense in Lip$_0(M,\mathbb R)$, then the unit ball of the Lipschitz-free space on $M$ is the closed convex hull of its strongly exposed points. Further, we prove that given a compact metric space $M$ which does not contain any isometric copy of $[0,1]$ and a Banach space $Y$, if SNA$(M,Y)$ is dense, then SNA$(M,Y)$ actually contains an open dense subset.



中文翻译:

获得 Lipschitz 映射的强范数密度的例子和应用

我们研究了这些 Lipschitz 映射的集合 SNA$(M,Y)$ 从(完全指向的)度量空间 $M$ 到强烈达到其范数的 Banach 空间 $Y$ 的密度(即,定义 Lipschitz norm 实际上是最大值)。我们提出了一些新的、有些违反直觉的例子,并给出了一些应用。

首先,我们证明对于任何 Banach 空间 $Y$,SNA$(\mathbb T,Y)$ 在 Lip$_0(\mathbb T,Y)$ 中都不稠密,其中 $\mathbb T$ 表示欧几里得平面。这提供了密度不成立的 Gromov 凹度量空间的第一个示例(即,每个分子都是 Lipschitz 自由空间的单位球的强烈暴露点)。

接下来,我们构造度量空间 $M$,满足 SNA$(M,Y)$ 在 Lip$_0(M,Y)$ 中是稠密的,不管 $Y$ 但包含 $[0,1]$ 的等距副本等等Lipschitz 自由空间 $\mathcal F(M)$ 不符合 Radon-Nikodym 性质,否定了 Cascales 等人提出的问题。在 2019 年和 Godefroy 在 2015 年。此外,可以产生这样一个 $M$ 的例子,它没有满足所有先前已知的强范数密度获得 Lipschitz 地图的充分条件。

最后,在其他应用中,我们证明如果 $M$ 是一个有界紧的度量空间,其中 SNA$(M,\mathbb R)$ 在 Lip$_0(M,\mathbb R)$ 中是稠密的,那么单位球$M$ 上的 Lipschitz-free 空间是其强暴露点的封闭凸包。此外,我们证明给定一个紧致度量空间 $M$ 不包含 $[0,1]$ 的任何等距副本和一个 Banach 空间 $Y$,如果 SNA$(M,Y)$ 是稠密的,那么 SNA $(M,Y)$ 实际上包含一个开放的密集子集。

更新日期:2021-02-01
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