James' theorem fails for starlike bodies

Starlike bodies are interesting in nonlinear functional analysis because th ey are strongly related to bump function sand to n-homogeneous polynomials on Banach spaces, and their geometrical proper ties are thus worth studying . In this paper we deal wit the question whether James' theorem on the char acterization of reflexivity holds for (smooth) starlike bodies, and we esta blish that a feeble form of this result is trivially true for starlike bodi es in nonreflexive Banach spaces, but a reasonable strong version of James' theorem for starlike bodies is never true, even in the smooth case. We als o study the related question as to how large the set of gradients of a bump function can be, and among other results we obtain the following new chara cterization of smoothness in Banach spaces: a Banach space X has a C-1 Lips chitz bump function if and only if there exists another C-1 smooth Lipschit z bump function whose set of gradients contains the unit ball of the dual s pace X*. This result might also be relevant to the problem of finding an As plund space with no smooth bump functions. (C) 2001 Academic Press.