The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces

Abstract

We prove the following new characterization of Cp Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a Cp smooth (Lipschitz)
bump function if and only if it has another Cp smooth (Lipschitz) bump function f such that its derivative does not vanish at any point in the interior of the support of f (that is, f does not satisfy Rolle's theorem). Moreover, the support of this bump can be assumed to be a smooth starlike body. The ``twisted tube'' method we use in the proof is interesting in itself, as it provides other useful characterizations of Cp smoothness related to the existence of a certain kind of deleting diffeomorphisms, as well as to the failure of Brouwer's fixed point theorem even for smooth self-mappings of starlike bodies in all infinite-dimensional spaces.