I-convexity and Q-convexity in Orlicz–Bochner function spaces equipped with the Luxemburg norm

Abstract

We study I-convexity and Q-convexity, two geometric properties introduced by Amir and Franchetti. We point out that a Banach space $X$ has the weak fixed-point property when $X$ is I-convex (or Q-convex) with a strongly bimonotone basis. By means of some characterizations of I-convexity and Q-convexity in Banach spaces, we obtain criteria for these two convexities in the Orlicz–Bochner function space $L_{(M)}(\mu ,X)$: that $L_{(M)}(\mu ,X)$ is I-convex (or Q-convex) if and only if $L_{(M)}(\mu )$ is reflexive and $X$ is I-convex (or Q-convex).