SMOOTH BUMP FUNCTIONS AND GEOMETRY OF BANACH SPACES R. DEVILLE, G. GODEFROY AND V. ZIZLER Abstract. Norms with moduli of smoothness of power type are constructed on spaces with the Radon-Nikodym property that admit pointwise Lipschitz bump functions with pointwise moduli of smoothness of power type. It is shown that no norms with pointwise moduli of rotundity of power type can exist on nonsuperreflexive spaces. A new smoothness characterization of spaces isomorphic to Hilbert spaces is given. §1. Introduction. In Banach space theory as well as in Analysis on Banach spaces, it is important to study families of smooth real valued functions with bounded and nonempty supports on given spaces (smooth bump functions) (cf. e.g., [BF], [Dev 2], [DGZ]). Such functions are usually constructed by composing smooth norms with appropriate real valued functions on the real line. R. Haydon has recently shown that there are Banach spaces which admit Lipschitz, continuously Frechet differentiable bump functions and admit no Gateaux differentiable norms ([HI], [H2], [H3]). In some classes of spaces and for some kinds of smoothness, the existence of a smooth bump function on X already implies the existence of a norm with a smoothness property not worse than that of the bump function. This is the case, for example, of separable spaces and Frechet smooth bump functions ([LW]) or of bump functions with locally uniformly continuous Frechet derivative on spaces X that do not contain an isomorphic copy of c 0 ([FWZ], [Fl], [DF]). In the first case, the result is obtained (rather indirectly) by using the tech- nique of rough norms and Kadec renorming of separable spaces by locally uniformly rotund norms ([Ka], [LW], cf. e.g., [DGZ]). In the second case,firsta bump function with uniformly continuous Frechet derivative is constructed on X, by using the Bessaga-Pelczynski characteriza- tion of spaces not containing isomorphic copies of c 0 . The norm is then con- structed by using level sets of a proper bump function on X. The latter method cannot, in general, work for nonuniformly differentiable bump functions, as shown in Example III. 10 below. The purpose of this paper is to present a new method for constructing norms with moduli of smoothness of power type from pointwise Lipschitz bump functions with pointwise moduli of smoothness of power type for spaces with the Radon-Nikodym property. Our technique shows that if a Banach space X has the Radon-Nikodym property and admits a Frechet differentiable bump function, then X admits a norm in which each closed convex and bounded [MATHEMATIKA, 40 (1993), 305-321]