  We prove that every separable infinite-dimensional Banach space admits a
G\^ateaux smooth and rotund norm which is not midpoint locally uniformly
rotund. Moreover, by using a similar technique, we provide in every
infinite-dimensional Banach space with separable dual a Fr\'echet smooth and
weakly uniformly rotund norm which is not midpoint locally uniformly rotund.
These two results provide a positive answer to some open problems by A. J.
Guirao, V. Montesinos, and V. Zizler.
