## $$I$$-convexity and $$Q$$-convexity in Orlicz-Bochner function spaces equipped with the Luxemburg norm.(English)Zbl 1421.46015

Summary: We study $$I$$-convexity and $$Q$$-convexity, two geometric properties introduced by D. Amir and C. Franchetti [Trans. Am. Math. Soc. 282, 275–291 (1984; Zbl 0543.46007)]. 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).

### MSC:

 46B20 Geometry and structure of normed linear spaces 46E40 Spaces of vector- and operator-valued functions 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Zbl 0543.46007
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### References:

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