The Vapnik-Chervonenkis dimension of cubes in ${\mathbb{R}}^d$

The Vapnik-Chervonenkis (VC) dimension of a collection of subsets of a set is an important combinatorial concept in settings such as discrete geometry and machine learning. In this paper we prove that the VC dimension of the family of $d$-dimensional cubes in ${\mathbb{R}}^d$ is $\lfloor (3d+1)\mathrm{/}2\rfloor$.