We study I-convexity and Q-convexity, two geometric properties introduced by Amir and Franchetti. We point out that a Banach space has the weak fixed-point property when 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 : that is I-convex (or Q-convex) if and only if is reflexive and is I-convex (or Q-convex).
"I-convexity and Q-convexity in Orlicz–Bochner function spaces equipped with the Luxemburg norm." Ann. Funct. Anal. 10 (1) 81 - 96, February 2019. https://doi.org/10.1215/20088752-2018-0010