Sharp isoperimetric inequalities on the Hamming cube near the critical exponent
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929424777084928 |
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| author | Durcik, Polona Ivanisvili, Paata Roos, Joris |
| author_facet | Durcik, Polona Ivanisvili, Paata Roos, Joris |
| contents | An isoperimetric inequality on the Hamming cube for exponents $β\ge 0.50057$ is proved, achieving equality on any subcube. This was previously known for $β\ge \log_2(3/2)\approx 0.585$. Improved bounds are also obtained at the critical exponent $β=0.5$, 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 Poincaré inequalities for Boolean-valued functions near $L^1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12674 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp isoperimetric inequalities on the Hamming cube near the critical exponent Durcik, Polona Ivanisvili, Paata Roos, Joris Classical Analysis and ODEs Combinatorics 46B09, 60E15, 05C35, 65G30 An isoperimetric inequality on the Hamming cube for exponents $β\ge 0.50057$ is proved, achieving equality on any subcube. This was previously known for $β\ge \log_2(3/2)\approx 0.585$. Improved bounds are also obtained at the critical exponent $β=0.5$, 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 Poincaré inequalities for Boolean-valued functions near $L^1$. |
| title | Sharp isoperimetric inequalities on the Hamming cube near the critical exponent |
| topic | Classical Analysis and ODEs Combinatorics 46B09, 60E15, 05C35, 65G30 |
| url | https://arxiv.org/abs/2407.12674 |