On a classical renorming construction of V. Klee

We further develop a classical geometric construction of V. Klee and show, typically, that if X is a nonreflexive Banach space with separable dual, then X admits an equivalent norm | ⋅ | which is Fréchet differentiable, locally uniformly rotund, its dual norm | ⋅ | ⁎ is uniformly Gâteaux differentiable, the weak⁎ and the norm topologies coincide on the sphere of ( X ⁎ , | ⋅ | ⁎ ) and, yet, | ⋅ | ⁎ is not rotund. This proves (a stronger form of) a conjecture of V. Klee.