Hurewiczʹs Theorems and Renormings of Banach Spaces

LetN(X) be the set of all equivalent norms on a separable Banach spaceX, equipped with the topology of uniform convergence on bounded subsets ofX. We show that ifXis infinite dimensional, the set of all locally uniformly rotund norms onXreduces every coanalytic set and, thus, is in particular non-Borel. Dually, we show the same result for the set of all continuously differentiable norms onX, under the assumptionX* is separable. This provides an analogue to a classical result of Mazurkiewicz within convex analysis.