Norm-one points in convex combinations of relatively weakly open subsets of the unit ball in the spaces $L_1(μ,X)$

TL;DR

The paper extends a known interiority property for finite convex combinations of relatively weakly open subsets from L_1(μ) to Lebesgue–Bochner spaces L_1(μ,X) when X is weakly uniformly rotund. It develops a suite of auxiliary lemmas, proves stability of property CWO-S under ℓ_1-sums, and then uses a finite-measure reduction plus careful partitioning and dual-estimate techniques to demonstrate that every finite convex combination of relatively weakly open subsets of B_{L_1(μ,X)} is weakly interior at its sphere points. The result shows that L_1(μ,X) inherits the CWO-S property from X under the wUR assumption, advancing the understanding of stability phenomena in Banach space geometry and Bochner spaces. The approach combines partitioning arguments, weak-topology neighborhood constructions, and wUR-based controls to achieve interiority, with potential implications for stability and geometry of related function spaces.

Abstract

In a paper published in 2020 in Studia Mathematica, Abrahamsen et al. proved that in the real space $L_1(μ)$, where $μ$ is a non-zero $σ$-finite (countably additive non-negative) measure, norm-one elements in finite convex combinations of relatively weakly open subsets of the unit ball are interior points of these convex combinations in the relative weak topology. In this paper that result is generalised by proving that the same is true in the (real or complex) Lebesgue--Bochner spaces $L_1(μ,X)$ where $X$ is a weakly uniformly rotund Banach space.

Theorems & Definitions (11)

• Definition : see ABHLP
• Theorem 1.1
• Proposition 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Proposition \ref{['prop: 1-sum of Banach spaces with property CWO-S has property CWO-S']}