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

Polona Durcik, Paata Ivanisvili, Joris Roos, Xinyuan Xie

Abstract

A sharp isoperimetric inequality for the Hamming cube is proved at the critical exponent $β=\frac12$. This follows up on previous work, where such bounds were established for $β$ near $\frac12$. 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.

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 . This follows up on previous work, where such bounds were established for near . As a consequence, this result settles a conjecture of Kahn and Park on cube partitions and yields a sharp 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.
Paper Structure (17 sections, 5 theorems, 94 equations)

This paper contains 17 sections, 5 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduction to a two-point inequality
  4. Properties of the function J
  5. Properties of the functions L and Q
  6. Computer-assisted proofs
  7. Proof of the two-point inequality
  8. Case J
  9. Case Q
  10. Case LJQ
  11. Case LJ
  12. Case QJQ
  13. Case QJ
  14. Near diagonal: $3/8\leq x\leq 1/2, 1/2\leq y\leq 5/8, x+y\geq 1$
  15. Far from diagonal: $\tfrac{1}{4}\leq x\leq \tfrac{1}{2}, \frac{5}{8}\leq y\le 1$, $x+y\ge 1$
  16. ...and 2 more sections

Key Result

Theorem 1.1

For all $A\subset \{0,1\}^n$ with $|A|\le \frac{1}{2}$,

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3: L
  • Lemma 2.4: Q