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.

We show a result slightly more general than the following. Let $K$ be a compact Hausdorff space, $F$ a closed subset of $K$ , and $d$ a lower semi-continuous metric on $K$ . Then each continuous function $f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on $K$ which is Lipschitz in $d$ . The extension has the same supremum norm and the same Lipschitz constant.

As a corollary we get that a Banach space $X$ is reflexive if and only if each bounded, weakly continuous and norm Lipschitz function defined on a weakly closed subset of $X$ admits a weakly continuous, norm Lipschitz extension defined on the entire space $X$ .

Let X be a separable infinite dimensional Banach space. There exist a closed set A ⊂ X which contains a translate of any compact set in the unit ball of X, and a bi-Lipschitz homeomorphism f of X onto X so that every line in X intersects f(A) in a set of Lebesgue measure zero.