Exercise 17, § 7

In the space $l^\infty(\mathbf{N})$, let A be the compact convex set defined in the exerc. 12 of II, p. 89 and let E be the closed vector subspace of $l^\infty(\mathbf{N})$ generated by A. Show that the lower bound of the distance between two parallel closed support hyperplanes of A in the space E is equal to 0, even though A is not contained in a closed hyperplane of E.