A topological characterization of dual strict convexity in Asplund spaces.

Let $X$ be an Asplund space. We show that the existence of an equivalent norm on $X$ having a strictly convex dual norm is equivalent to the dual unit sphere $S_{X^*}$ (equivalently $X^*$) possessing a non-linear topological property called ($*$), which was introduced by J. Orihuela, S. Troyanski and the author.