On the Isoperimetric Profile of the Hypercube

2023/05/31 by Federico Glaudo, Glaudo, Federico · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Metric Geometry (math.MG) #Point processes and geometric inequalities #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2305.20051

openalex publication_date 2023/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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→∞.

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