Convexity and Haar null sets

It is shown that for every closed, convex and nowhere dense subset $C$ of a superreflexive Banach space $X$ there exists a Radon probability measure $\mu$ on $X$ so that $\mu (C+x)=0$ for all $x\in X$. In particular, closed, convex, nowhere dense sets in separable superreflexive Banach spaces are Haar null. This is unlike the situation in separable nonreflexive Banach spaces, where there always exists a closed convex nowhere dense set which is not Haar null.