Diametrically complete sets and normal structure

We prove that in some classes of reflexive Banach spaces every maximal diametral set must be diametrically complete, thus showing that diametrically complete sets may have empty interior also in reflexive spaces. As a consequence, we prove that, in those spaces, normal structure is equivalent to the weaker property that, for every bounded set, the absolute Chebyshev radius is strictly smaller than the diameter. Moreover,we prove that, in any normed space, the two classes of diametrically complete sets and of sets of constant radius from the boundary coincide if the space has normal structure.