Unbounded nested sequences of balls in Banach spaces

It is not so difficult to prove that any closed convex solid cone in a Banach space is the union of an unbounded nested sequence of balls in some equivalent norm. In this paper the converse question is considered : namely, under what conditions is the union of an unbounded nested sequence of balls a cone? It was proved in joint work of the present authors, P. Bandyopadhyay, and M. Martin [Houston J. Math. 29 (2003), 173-193] that in finite dimensions such a union is always a cone. In this paper we reveal the following infinite-dimensional phenomena : it is possible that the union is a cone but no vertex of it belongs to the (closed) subspace generated by the centers of the balls. Thus an answer to the question above in general is "no". By using the concept of a norming cone, we establish necessary and sufficient conditions for the union of an unbounded nested sequence of balls to be a cone. In this context we get necessary and sufficient conditions for a Lindenstrauss space to be polyhedral, and treat also the space of affine continuous functions on a compact metrizable simplex. Some other related problems are considered.