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

On the Vapnik-Chervonenkis dimension of products of intervals in $\mathbb{R}^d$