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Journal of Convex Analysis 14 (2007), No. 2, 249--269
Copyright Heldermann Verlag 2007

High Order Smoothness and Asymptotic Structure in Banach Spaces

R. Gonzalo
Dep. de Matemática Aplicada, Facultad de Informática, Universidad Politécnica, Montegancedo / Boadilla del Monte, 28660 Madrid, Spain

J. A. Jaramillo
Dep. de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

S. L. Troyanski
Dep. de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo - Murcia, Spain
Permanent Address: Inst. of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str. block 8, 1113 Sofia, Bulgaria

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.

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