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

Abstract

A sharp isoperimetric inequality for the Hamming cube is proved at the critical exponent ${\hat{I}}^2=\frac{1}{2}$. This follows up on previous work, where such bounds were established for ${\hat{I}}^2$ near $\frac{1}{2}$. As a consequence, this result settles a conjecture of Kahn and Park on cube partitions and yields a sharp $L^1$ Poincaré 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.

14 pages; accompanying code at https://github.com/roos-j/dir24-isoperim