TRANSLATING FINITE SETS INTO CONVEX SETS

Source abstract

Let X be a reflexive Banach space, and let C ⊂ X be a closed, convex and bounded set with empty interior. Then, for every δ > 0, there is a nonempty finite set F ⊂ X with an arbitrarily small diameter, such that C contains at most δ · | F | points of any translation of F . As a corollary, a separable Banach space X is reflexive if and only if every closed convex subset of X with empty interior is Haar null.