The Generalised (Uniform) Mazur Intersection Property

TL;DR

This paper introduces and characterizes stronger and uniform versions of the generalized Mazur Intersection Property in Banach spaces, providing a more comprehensive framework and analogues of classical characterizations.

Contribution

It defines stronger and uniform variants of the generalized MIP and establishes their characterizations, enriching the theoretical understanding of intersection properties in Banach spaces.

Findings

Introduction of stronger $ ext{C}$-MIP and $ ext{C}$-UMIP

Complete analogues of classical MIP characterizations

Rich characterizations of the strong uniform $ ext{C}$-UMIP

Abstract

Given a family $\mathcal{C}$ of closed bounded convex sets in a Banach space $X$, we say that $X$ has the $\mathcal{C}$-MIP if every $C \in \mathcal{C}$ is the intersection of the closed balls containing it. In this paper, we introduce a stronger version of the $\mathcal{C}$-MIP and show that it is a more satisfactory generalisation of the MIP inasmuch as one can obtain complete analogues of various characterisations of the MIP. We also introduce uniform versions of the (strong) $\mathcal{C}$-MIP and characterise them analogously. Even in this case, the strong $\mathcal{C}$-UMIP appears to have richer characterisations than the $\mathcal{C}$-UMIP.