Hurewicz's theorems and renormings of Banach spaces

Let N(X) be the set of all equivalent norms on a separable Banach space X, equipped with the topology of uniform convergence on bounded subsets of X. We show that if X is infinite dimensional, the set of all locally uniformly rotund norms on X reduces every coanalytic set and, thus, is in particular non-Borel. Dually, we show the same result for the set of all continuously differentiable norms on X, under the assumption X* is separable. This provides an analogue to a classical result of Mazurkiewicz within convex analysis. (C) 1996 Academic Press, Inc.