Wanzhong Gong, Xiaoli Dong, Kangji Wang. Source: Annals of Functional Analysis, Volume 10, Number 1, 81--96.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).