Smoothness and Rotundity in Banach spaces

The concept of rotundity is not far from differentiability. Some paper investigate the relation between rotundity and smoothness. In this paper we will explain some new relation between rotundity and very smoothness. A Banach space is rotund if the midpoint of every two distinct points of unit sphere is in the open unit ball of Banach space. A Banach space 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 norm of Banach space and the norm of second dual of Banach space are Gateaux differentiable at every non zero point of Banach space.