2024/07/17 by Polona Durcik, Paata Ivanisvili, Durcik, Polona +3 · 1 citation
Mathematics · #05C35 #46B09 #60E15 #65G30 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Point processes and geometric inequalities #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2407.12674
openalex publication_date 2024/07/17 · openalex created_date 2024/10/26 · openalex updated_date 2026/07/28
An isoperimetric inequality on the Hamming cube for exponents β≥ 0.50057 is proved, achieving equality on any subcube. This was previously known for β≥ log2(3/2)≈ 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 L1.