Diametral points and diametral pairs in Banach spaces

Abstract

We prove that every infinite dimensional Banach space contains symmetric, bounded, closed, convex sets with nonempty interior that have no diametral points. For diametral pairs, we show that, in a reflexive Banach space, a point in the boundary of a set of constant width C belongs to a diametral pair if and only if it is a support point of C. Therefore, if C has nonempty interior, all its boundary points belong to diametral pairs. We prove that, in some reflexive space, a set of constant width (with empty interior) does exist whose boundary contains a dense subset of points that do not belong to any diametral pair. Moreover we prove that having the whole boundary consisting of diametral pairs does not characterize constant width sets in the family of all bounded, closed, convex sets with nonempty interior of a reflexive space.