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arXiv · 2602.20462

Sharp isoperimetric inequalities on the Hamming cube II: The critical exponent

Polona Durcik·Paata Ivanisvili·Joris Roos·Xinyuan Xie
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Abstract

A sharp isoperimetric inequality for the Hamming cube is proved at the critical exponent $\beta=\frac12$. This follows up on previous work, where such bounds were established for $\beta$ near $\frac12$. As a consequence, this result settles a conjecture of Kahn and Park on cube partitions and yields a sharp $L^1$ Poincar\'e inequality for Boolean-valued functions. It also confirms a low-noise limit for balanced functions predicted by the Hellinger conjecture on noisy Boolean channels in information theory.

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Polona Durcik, Paata Ivanisvili, Joris Roos, Xinyuan Xie. 2026-02-24. Sharp isoperimetric inequalities on the Hamming cube II: The critical exponent. https://arxiv.org/abs/2602.20462

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