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.