  In $\mathbb{R}^d$, a closed, convex set has zero Lebesgue measure if and only
its interior is empty. More generally, in separable, reflexive Banach spaces,
closed and convex sets are Haar null if and only if their interior is empty. We
extend this facts by showing that a closed, convex set in a separable Banach
space is Haar null if and only if its weak$^*$ closure in the second dual has
empty interior with respect to the norm topology. It then follows that, in the
metric space of all nonempty, closed, convex and bounded subsets of a separable
Banach space, converging sequences of Haar null sets have Haar null limits.

Functional Analysis