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We provide, in every infinite-dimensional separable Banach space, an averagelocally uniformly rotund (and hence rotund) Gateaux smooth renorming which isnot locally uniformly rotund. This solves an open problem posed by A.J. Guirao,V. Montesinos, and V. Zizler.

Rotund Gateaux smooth norms which are not locally uniformly rotund