The failure of Rolle's theorem in infinite-dimensional Banach spaces

We prove the following new characterization of C-P (Lipschitz) smoothness i n Banach spaces. An infinite-dimensional Banach space X has a C-P smooth (L ipschitz) bump function if and on IS if it has another C-P smooth (Lipschit z) bump function f such that its derivative dues not vanish at any point in the interior off the support of f (that is. f does not satisfy Rolle's the orem). Moreover, the support of this hump can be assumed to be a smooth sta rlike body. The "twisted tube" method we use in the proof is interesting in itself, as it provides other useful characterizations of C-P smoothness re lated to the existence of a certain kind of deleting diffeomorphisms, as we ll as to the failure of Brouwer's fixed point theorem even for smooth self- mappings of starlike bodies in all infinite-dimensional spaces. (C) 2001 Ac ademic Press.