Convex sets without diametral pairs

Let X be an infinite dimensional normed linear space. It is not difficult to see that arbitrarily near (in the Hausdorff metric) to the unit ball of X there exists a nonempty closed convex set whose diameter is not attained. We show that such sets are dense in the metric space of all nonempty bounded closed convex subsets of X if and only if either X is not a reflexive Banach space or X is a reflexive Banach space in which every weakly closed set contained in the unit sphere S_X has empty relative interior in S_X.