On uniform Mazur intersection property

TL;DR

This paper characterizes Banach spaces with the Uniform Mazur Intersection Property (UMIP) through the behavior of functionals in the dual space, linking geometric properties to functional-analytic conditions.

Contribution

It establishes a new equivalence between UMIP and the uniform w*-semidenting points of the dual unit sphere, extending understanding of geometric properties in Banach spaces.

Findings

UMIP characterized by dual space functional behavior

Equivalent conditions for uniform w*-MIP established

New insights into Banach space geometry obtained

Abstract

In this paper, we show that a Banach space $X$ has the Uniform Mazur Intersection Property (UMIP) if and only if every $f \in S(X^*)$ is uniformly w*-semidenting point of $B(X^*)$. We also prove analogous results for uniform w*-MIP.