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/

Global: Zendy Plus annual

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

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We show that a Banach space with numerical index one cannot enjoy goodconvexity or smoothness properties unless it is one-dimensional. For instance,it has no WLUR points in its unit ball, its norm is not Frechet smooth and itsdual norm is neither smooth nor strictly convex. Actually, these results alsohold if the space has the (strictly weaker) alternative Daugavet property. Weconstruct a (non-complete) strictly convex predual of an infinite-dimensional$L_1$ space (which satisfies a property called lushness which implies numericalindex~1). On the other hand, we show that a lush real Banach space is neitherstrictly convex nor smooth, unless it is one-dimensional. In particular, if asubspace $X$ of the real space $C[0,1]$ is smooth or strictly convex, then$C[0,1]/X$ contains a copy of $C[0,1]$. Finally, we prove that the dual of anylush infinite-dimensional real space contains a copy of $\ell_1$.

Convexity and smoothness of Banach spaces with numerical index one