Structure of nested sequences of balls in Banach spaces

In this paper, we study the structure of the union of unbounded nested sequences of balls, and use them to characterize some geometric properties of X*. We show that the union of an unbounded nested sequence of balls is a cone if the centers of the balls lie in a finite dimensional subspace. However, in general, such a union need not be a cone. In fact, examples can be constructed, up to renorming, in any infinite dimensional Banach space. We also study when such an union is the intersection of at most k half-spaces, and relate it with the number of extreme points of any face of the dual ball.