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

TL;DR

This paper proves a sharp isoperimetric inequality on the Hamming cube at the critical exponent, confirming conjectures in combinatorics and information theory, and establishing precise bounds for Boolean functions.

Contribution

It establishes the first sharp isoperimetric inequality at the critical exponent on the Hamming cube, resolving a conjecture and connecting to information theory.

Findings

Proves sharp isoperimetric inequality at critical exponent $eta=1/2$

Settles a conjecture of Kahn and Park on cube partitions

Confirms a low-noise limit for balanced Boolean functions

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.