On the Isoperimetric Profile of the Hypercube

We prove that a subset of the hypercube $(0,1{)}^d$ with volume sufficiently close to $\frac{1}{2}$ has (relative) perimeter greater than or equal to $1$. This settles a conjecture by Brezis and Bruckstein. We also prove that, in contrast with what happens for the high-dimensional sphere ${\mathbb{S}}^d$, the isoperimetric profile of the hypercube $(0,1{)}^d$ does not converge to the Gaussian isoperimetric profile as $d\to \mathrm{\infty }$.