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

2026-02-24

AI Summary

This work establishes a sharp isoperimetric inequality on the Hamming cube at the critical exponent β = 1/2, thereby resolving the long-standing Kahn–Kalai (formerly Kahn–Park) cube partition conjecture and confirming the prediction of the information-theoretic Hellinger conjecture for low-noise Boolean channels. By integrating harmonic analysis, analysis of Boolean functions, and probabilistic methods—and leveraging the duality between isoperimetric and Poincaré inequalities—the study provides the first complete characterization at this critical regime. The results unify several conjectures across combinatorics, probability, and information theory, and yield a sharp L¹ Poincaré inequality for Boolean functions, significantly advancing the understanding of high-dimensional discrete geometry and noise stability theory.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

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.

Problem

• isoperimetric inequality
• Hamming cube
• critical exponent
• Boolean functions
• Poincaré inequality

Innovation

• isoperimetric inequality
• Hamming cube
• critical exponent
• Boolean functions
• Poincaré inequality