English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We study the density of the set $\operatorname{SNA}(M,Y)$ of those Lipschitzmaps from a (complete pointed) metric space $M$ to a Banach space $Y$ whichstrongly attain their norm (i.e.\ the supremum defining the Lipschitz norm isactually a maximum). We present new and somehow counterintuitive examples, andwe give some applications. First, we show that $\operatorname{SNA}(\mathbbT,Y)$ is not dense in ${\mathrm{Lip}}_0(\mathbb T,Y)$ for any Banach space $Y$,where $\mathbb T$ denotes the unit circle in the Euclidean plane. This providesthe first example of a Gromov concave metric space (i.e.\ every molecule is astrongly exposed point of the unit ball of the Lipschitz-free space) for whichthe density does not hold. Next, we construct metric spaces $M$ satisfying that$\operatorname{SNA}(M,Y)$ is dense in ${\mathrm{Lip}}_0(M,Y)$ regardless $Y$but which contains an isometric copy of $[0,1]$ and so the Lipschitz-free space$\mathcal F(M)$ fails the Radon--Nikod\'{y}m property, answering in thenegative a posed question. Furthermore, an example $M$ can be produced failingall the previously known sufficient conditions to get the density of stronglynorm attaining Lipschitz maps. Finally, among other applications, we prove thatgiven a compact metric $M$ which does not contains any isometric copy of$[0,1]$ and a Banach space $Y$, if $\operatorname{SNA}(M,Y)$ is dense, then$\operatorname{SNA}(M,Y)$ actually contains an open dense subset and$B_{\mathcal F(M)}=\overline{{\mathrm{co}}}(\operatorname{str-exp}(B_{\mathcalF(M)}))$. Further, we show that if $M$ is a boundedly compact metric space forwhich $\operatorname{SNA}(M,\mathbb R)$ is dense in ${\mathrm{Lip}}_0(M,\mathbbR)$, then the unit ball of the Lipschitz-free space on $M$ is the closed convexhull of its strongly exposed points.

Examples and applications of the density of strongly norm attaining Lipschitz maps