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 Tools annual

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 Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Later, R. R. Phelps [11] provided a dual characterization of this property for finite-dimensional spaces. Nearly two decades later, Phelps’ results were extended by J. R. Giles, D. A. Gregory and B. Sims [9] to general normed linear spaces. They also showed that in dual Banach spaces the MIP implies reflexivity, and considered the weaker property that every weak* compact convex set in a dual space is the intersection of balls (Property weak*-I). Subsequently, there appeared several papers dealing with similar intersection properties for compact convex sets [14, 12], weakly compact convex sets [16] and compact convex sets with finite affine dimension [13]. In the present work, we give a unified treatment of the intersection properties for these diverse classes of sets by considering the MIP for the members of a general family of closed bounded convex sets in a Banach space, and show that all the known results follow as special cases of our result. We also introduce a new condition of separation of convex sets which turns out to be equivalent to the intersection property in all known cases. This strengthens the results of Zizler [15]. As another application of our result, we extend our previous work [1] on lifting the MIP from a Banach space X to the Lebesgue–Bochner space L(μ,X) (1 < p < ∞). We should point out that our proofs are usually modifications, refinements and adaptations to our very general set-up of arguments for particular cases to be found in [9], [12] and [14].

The Mazur intersection property for families of closed bounded convex sets in Banach spaces