  We study combinatorial complexity of certain classes of products of intervals
in $\mathbb{R}^d$, from the point of view of Vapnik-Chervonenkis geometry. As a
consequence of the obtained results, we conclude that the Vapnik-Chervonenkis
dimension of the set of balls in $\ell_\infty^d$ -- which denotes $\R^d$
equipped with the sup norm -- equals $\lfloor (3d+1)/2\rfloor$.
