Diameter 2 properties and convexity

TL;DR

This paper constructs specific MLUR Banach spaces with diameter 2 properties and convexity features, providing new insights into the geometry of Banach spaces and their projections.

Contribution

It introduces a new MLUR renorming of $C[0,1]$ with unique projection properties and constructs spaces with diameter 2 and convexity properties.

Findings

Existence of MLUR spaces with D2P and LD2P+ properties.

Construction of an MLUR space with convex combinations of slices of small diameter.

Characterization of weakly compact projections satisfying a specific norm equation.

Abstract

We present an equivalent midpoint locally uniformly rotund (MLUR) renorming $X$ of $C[0,1]$ on which every weakly compact projection $P$ satisfies the equation $\|I-P\| = 1+\|P\|$ ($I$ is the identity operator on $X$). As a consequence we obtain an MLUR space $X$ with the properties D2P, that every non-empty relatively weakly open subset of its unit ball $B_X$ has diameter 2, and the LD2P+, that for every slice of $B_X$ and every norm 1 element $x$ inside the slice there is another element $y$ inside the slice of distance as close to 2 from $x$ as desired. An example of an MLUR space with the D2P, the LD2P+, and with convex combinations of slices of arbitrary small diameter is also given.