Geometrical implications of the existence of very smooth bump functions in Banach spaces (Q582528)

The norm on any \(L_ p\)-space is a \(C^{\infty}\)-function iff p is an even integer. Motivated by results of this type, the author proves a converse type result: If there is a non-zero real-valued \(C^{\infty}\)- smooth function on a Banach space X with bounded support, X is of exact cotype 2k with \(X\supseteq \ell^{2k}\) for some \(k\in {\mathbb{N}}\) or X contains a copy of \(c_ 0\). Moreover, if on X there is a non-zero \(C^ 4\)-smooth function with bounded support and if X is of cotype \(q<4\), then X is isomorphic to a Hilbert space.