Papers › Sharp isoperimetric inequalities on the Hamming cube near the critical exponent
Sharp isoperimetric inequalities on the Hamming cube near the critical exponent
Polona Durcik, Paata Ivanisvili, Joris Roos
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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 Poincar\'e inequalities for Boolean-valued functions near L¹.
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