A class of P-convex spaces lacking normal structure

We consider a family of renormings of the infinite-dimensional separable Hilbert space introduced in literature by R.C. James in the sixties and prove that all its members are P-convex. It is known that these spaces lack normal structure when they are sufficiently far, in the sense of the Banach-Mazur distance, from being Hilbert, hence our result provides the first example of a P-convex space without normal structure.