\(I\)-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).


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
Full Text: DOI Euclid


[1] G. Alherk and H. Hudzik, Uniformly non-\(l^{(1)}_{n}\) Musielak–Orlicz spaces of Bochner type, Forum Math. 1 (1989), no. 4, 403–410. · Zbl 0686.46016
[2] D. Amir and C. Franchetti, The radius ratio and convexity properties in normed linear spaces, Trans. Amer. Math. Soc. 282, no. 1 (1984), 275–291. · Zbl 0543.46007
[3] A. Beck, A convexity condition in Banach spaces and the strong law of large numbers, Proc. Amer. Math. Soc. 13 (1962), no. 2, 329–334. · Zbl 0108.31401
[4] S. Chen, Geometry of Orlicz Spaces, Dissertationes Math. (Rozprawy Mat.) 356, Polish Acad. Sci., Warsaw, 1996. · Zbl 1089.46500
[5] J. García-Falset, E. Llorens-Fuster, and E. M. Mazcuñán-Navarro, The fixed point property and normal structure for some B-convex Banach spaces, Bull. Aust. Math. Soc. 63 (2001), no. 1, 75–81. · Zbl 0986.47047
[6] J. García-Falset, E. Llorens-Fuster, and E. M. Mazcuñán-Navarro, Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings, J. Funct. Anal. 233 (2006), no. 2, 494–514. · Zbl 1120.46006
[7] S.-Z. Huang and J. M. A. M. Neerven, B-Convexity, the analytic Radon–Nikodym property, and individual stability of \(C_{0}\)-semigroups, J. Math. Anal. Appl. 231 (1999), no. 1, 1–20. · Zbl 0943.47029
[8] H. Hudzik, Some class of uniformly nonsquare Orlicz–Bochner spaces, Comment. Math. Univ. Carolin. 26 (1985), no. 2, 269–274. · Zbl 0579.46022
[9] H. Hudzik, Uniformly non-\(l_{n}^{(1)}\) Orlicz spaces with Luxemburg norm, Studia Math. 81 (1985), no. 3, 271–284. · Zbl 0591.46018
[10] R. C. James, Uniformly non-square Banach spaces, Ann. of Math. (2) 80 (1964), no. 3, 542–550. · Zbl 0132.08902
[11] A. Kamínska and B. Turett, Uniformly non-\(l_{n}^{(1)}\) Orlicz–Bochner spaces, Bull. Pol. Acad. Sci. Math. 35 (1987), no. 3–4, 211–218. · Zbl 0631.46022
[12] P. Kolwicz and R. Pluciennik, P-convexity of Orlicz–Bochner spaces, Proc. Amer. Math. Soc. 126 (1998), no. 8, 2315–2322. · Zbl 0896.46019
[13] C. A. Kottman, Packing and reflexivity in Banach spaces, Trans. Amer. Math. Soc. 150, no. 2 (1970), 565–576. · Zbl 0208.37503
[14] S. V. R. Naidu and K. P. R. Sastry, Convexity conditions in normed linear spaces, J. Reine Angew. Math. 297 (1978), no. 1, 35–53. · Zbl 0364.46009
[15] H. F. Nathansky and E. Llorens-Fuster, Comparison of P-convexity, O-convexity and other geometrical properties, J. Math. Anal. Appl. 396 (2012), no. 2, 749–758. · Zbl 1270.46014
[16] M. A. Smith and B. Turett, Rotundity in Lebesgue–Bochner function spaces, Trans. Amer. Math. Soc. 257, no. 1 (1980), 105–118. · Zbl 0368.46039
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