High order smoothness and asymptotic structure in banach spaces

In this paper 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 smoothness of Nakano sequence spaces.