LFC bumps on separable Banach spaces (Q847780)

Let \(X\) be a Banach space. A function \(b:X\to \mathbb{R}\) is a bump function if it has bounded and nonempty support. The function \(b\) depends locally on finitely many coordinates (LFC for short) if for each \(x\in X\) there are a neighborhood \(U\) of \(x\) and a finite subset \(F\subset X^*\) such that \(b(y)=b(z)\) whenever \(y,z\in U\) such that \(f(y)=f(z)\) for all \(f\in F\). A norm is said to be LFC if it is LFC away from the origin. The LFC notion is closely related to the class of polyhedral Banach spaces. \textit{P. Hájek} [Proc. Am. Math. Soc. 123, 3817--3821 (1995; Zbl 0853.46010)] characterized them as those admitting an equivalent \(\mathcal{C}^{\infty}\)-smooth and LFC norm. It remains an open problem whether every separable Banach space with a \(\mathcal{C}^{\infty}\)-smooth LFC bump is polyhedral. In the paper under review, the authors construct a \(\mathcal{C}^{\infty}\)-smooth LFC bump function in every separable Banach space admitting a continuous, LFC bump function. This result answers a problem posed by several authors (see, e.g., [\textit{P. Hájek, M. Johanis}, J. Math. Ann. Appl. 338, 1131--1139 (2008; Zbl 1136.46005)]).