A topological characterization of dual strict convexity in Asplund spaces

Abstract

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 $S_{X^{\ast }}$ (equivalently $X^{\ast }$) possessing a non-linear topological property called ($\ast$), which was introduced by J. Orihuela, S. Troyanski and the author.