Geodesic
High Order Smoothness and Asymptotic Structure in Banach Spaces
Journal of convex analysis, Tome 14 (2007) no. 2, pp. 249-269

Voir la notice de l'article provenant de la source Heldermann Verlag

We study the connections between moduli of asymptotic convexity and smoothness of a Banach space, and the existence of high order differentiable bump functions or equivalent norms on the space. The existence of a high order uniformly differentiable bump function is related to an asymptotically uniformly smooth renorming of power type. On the other hand, the asymptotic uniform convexity of power type is related to the existence of high order rough norms. Finally, we also obtain some applications to the best order smoothness of Nakano sequence spaces. 
@article{JCA_2007_14_2_JCA_2007_14_2_a2,
     author = {R. Gonzalo and J. A. Jaramillo and S. L. Troyanski},
     title = {High {Order} {Smoothness} and {Asymptotic} {Structure} in {Banach} {Spaces}},
     journal = {Journal of convex analysis},
     pages = {249--269},
     publisher = {mathdoc},
     volume = {14},
     number = {2},
     year = {2007},
     url = {http://geodesic.mathdoc.fr/item/JCA_2007_14_2_JCA_2007_14_2_a2/}
}
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R. Gonzalo; J. A. Jaramillo; S. L. Troyanski. High Order Smoothness and Asymptotic Structure in Banach Spaces. Journal of convex analysis, Tome 14 (2007) no. 2, pp. 249-269. http://geodesic.mathdoc.fr/item/JCA_2007_14_2_JCA_2007_14_2_a2/