Rotund Gâteaux smooth norms which are not locally uniformly rotund

Abstract

We provide, in every infinite-dimensional separable Banach space, an average locally uniformly rotund (and hence rotund) Gâteaux smooth renorming which is not locally uniformly rotund. This solves an open problem posed in a recent monograph by A.J. Guirao, V. Montesinos, and V. Zizler.