Smoothness and Rotundity in Banach Spaces

The concept of rotundity is not far from differentiability. there are several papers in the literature devoted to the study of relations between rotundity and smoothness in banach spaces. in this paper, we study new relations between some kinds of rotundity and smoothness in banach spaces. in particular, we investigate relations between one kind of rotundity, which is called strongly very rotund, and very smoothness, in banach spaces. a banach space is rotund if the midpoint of every two distinct points of unit sphere is in the open unit ball of the banach space. a banach space x is smooth if its norm is gateaux differentiable at every non-zero point of the space and it is very smooth if the norm is very gateaux differentiable, that is, the second dual norm in the second dual of x is gateaux differentiable at every non zero point of x ?.