New Families of Convex Sets Related to Diametral Maximality

Eggleston proved in a landmark monograph that, in every finite dimensional normed space, a bounded closed convex set with constant radius from its boundary is diametrically maximal. We show that this is no longer true in general and we characterize a set with constant radius by means of an equation involving its radius and diameter. A somewhat similar equation yields the definition of a constant difference set, a notion which turns out to be stronger than diametrically maximal but weaker than constant width. We investigate the interplay of these notions with the geometry of the underlying Banach space.