Geodesic
Rotund Renormings in Spaces of Bochner Integrable Functions
Journal of convex analysis, Tome 22 (2015) no. 4, pp. 1025-1039
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We show that if μ is a probability measure and X is a Banach space, then the Lebesgue-Bochner space L1(μ,X) admits an equivalent norm which is rotund (uniformly rotund in every direction, locally uniformly rotund, or midpoint locally uniformly rotund) if X does. We also prove that if X admits a uniformly rotund norm, then the space L1(μ,X) has an equivalent norm whose restriction to every reflexive subspace is uniformly rotund. This is done via the Luxemburg norm associated to a suitable Orlicz function.
Classification : 46B03, 46B20, 46E40
Mots-clés : Lebesgue-Bochner space, rotund norm, URED norm, LUR norm, MLUR norm, UR norm, Luxemburg norm, Orlicz function 
@article{JCA_2015_22_4_JCA_2015_22_4_a6,
     author = {M. Fabian and S. Lajara},
     title = {Rotund {Renormings} in {Spaces} of {Bochner} {Integrable} {Functions}},
     journal = {Journal of convex analysis},
     pages = {1025--1039},
     year = {2015},
     volume = {22},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JCA_2015_22_4_JCA_2015_22_4_a6/}
}
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M. Fabian; S. Lajara. Rotund Renormings in Spaces of Bochner Integrable Functions. Journal of convex analysis, Tome 22 (2015) no. 4, pp. 1025-1039. http://geodesic.mathdoc.fr/item/JCA_2015_22_4_JCA_2015_22_4_a6/