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.