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arXiv · 2305.20051

On the Isoperimetric Profile of the Hypercube

Federico Glaudo
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Abstract

We prove that a subset of the hypercube $(0,1)^d$ with volume sufficiently close to $\frac12$ 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 $d\to\infty$.

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Federico Glaudo. 2023-05-31. On the Isoperimetric Profile of the Hypercube. https://arxiv.org/abs/2305.20051

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