{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/sharp-isoperimetric-inequalities-on-the","title":"Sharp isoperimetric inequalities on the Hamming cube near the critical exponent","arxiv_id":"2407.12674","date":"2024-07-17","proceeding":null,"authors":["Polona Durcik","Paata Ivanisvili","Joris Roos"],"abstract":"An isoperimetric inequality on the Hamming cube for exponents $\\beta\\ge 0.50057$ is proved, achieving equality on any subcube. This was previously known for $\\beta\\ge \\log_2(3/2)\\approx 0.585$. Improved bounds are also obtained at the critical exponent $\\beta=0.5$, including a bound that is asymptotically sharp for small subsets. A key ingredient is a new Bellman-type function involving the Gaussian isoperimetric profile which appears to be a good approximation of the true envelope function. Verification uses computer-assisted proofs and interval arithmetic. Applications include progress towards a conjecture of Kahn and Park as well as sharp Poincar\\'e inequalities for Boolean-valued functions near $L^1$.","url_abs":"https://arxiv.org/abs/2407.12674v1","url_pdf":"https://arxiv.org/pdf/2407.12674v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"sharp-isoperimetric-inequalities-on-the","repo_url":"https://github.com/roos-j/dir24-isoperim","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}