Differentiability and ball-covering property in Banach spaces

In this paper, author proves that if X1 and X2 are Gâteaux differentiable space, then X1 and X2 have the ball-covering property if and only if (X1×X2,‖⋅‖p) and (X1×X2,‖⋅‖∞) have the ball-covering property, where ‖(x,y)‖p=(‖x‖1p+‖y‖2p)1p, p∈[1,+∞) and ‖(x,y)‖∞=max⁡{‖x‖,‖y‖}.