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

Abstract

In a paper published in 2020 in Studia Mathematica, Abrahamsen et al. proved that in the real space $L_1(\mu )$, where $\mu$ is a non-zero $\sigma$-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 $L_1(\mu ,X)$ where $X$ is a weakly uniformly rotund Banach space.