Walk the dog, or: products of open balls in the space of continuous functions

Abstract

Let $C[0,1]$ be the Banach algebra of real valued continuous functions on $[0,1]$, provided with the supremum norm. For $f,g\in C[0,1]$ and balls $B_{f}$, $B_{g}$ with center $f$ and $g$, respectively, it is not necessarily true that $f\cdot g$ is in the interior of $B_{f}\cdot B_{g}$. In the present paper we characterize those pairs $f,g$ where this is the case. The problem is illustrated by using a suitable translation. One studies walks in a landscape with hills and valleys where an accompanying dog can move in a certain prescribed way.