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 $β=\frac12$. 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.

• The paper establishes a sharp isoperimetric inequality on the Boolean hypercube at the critical exponent β = 1/2, achieving equality for subcubes.
• The methodology leverages advanced Bellman function techniques and rigorous computer-assisted interval arithmetic to partition and verify the inequality across multiple cases.
• The results resolve the Kahn-Park cube-partition conjecture and improve L¹ Poincaré inequalities for Boolean functions, with impactful implications for information theory.

Sharp Isoperimetric Inequalities on the Hamming Cube at the Critical Exponent

Introduction and Main Contributions

The paper "Sharp isoperimetric inequalities on the Hamming cube II: The critical exponent" (2602.20462) establishes the sharp isoperimetric inequality on the Boolean hypercube $\{0,1\}^n$ at the critical exponent $\beta=\frac{1}{2}$. This critical endpoint addresses a question at the intersection of discrete isoperimetry and concentration of measure, solidifying a conjecture by Kahn and Park concerning partitions of the cube and the fundamental behavior of boundaries associated with sets of given measure under the Hamming metric.

The primary result is that for all subsets $A \subset \{0,1\}^n$ of size $|A| \leq 1/2$, the inequality

$\mathbf{E} \sqrt{h_A} \ge |A| \sqrt{\log_2(1/|A|)}$

holds, where $h_A(x)$ is the vertex degree of $x \in A$ into $A^c$. Further, equality is achieved specifically for subcubes, highlighting the sharpness of the bound. The work leverages refined Bellman function methods in conjunction with rigorous computer-assisted inequalities.

Theoretical implications extend to Poincaré-type inequalities for Boolean functions, optimality in problems of Boolean function sensitivity, and confirmation of information-theoretic predictions regarding noisy Boolean channels, specifically a low-noise limit case of the Hellinger conjecture.

Technical Approach

Reduction to Two-Point Inequalities and Bellman Function Construction

Following a modern program initiated by Kahn and Park, the discrete isoperimetric problem is reduced to verifying an explicit two-point inequality for functions $B:[0,1]\to [0,\infty)$ with specific boundary and interpolation constraints. This recasts the isoperimetric statement as:

$$G[B](x,y) \coloneqq \max\left\{ G_1[B](x,y), G_2[B](x,y) \right\} \geq 0 \quad \forall\, 0\leq x \leq y \leq 1,$$

for carefully constructed $G_1$, $G_2$, related to the Bellman function formalism. The main technical step is the design and verification of a Bellman function $B_{w_1}$, piecewise smooth, interpolating between $L(x)=x\sqrt{\log_2(1/x)}$ at small $x$, an explicit interpolating cubic $Q(x)$, and a Gaussian isoperimetric profile scaling ${\mathrm{J}_w(x)}$ for $x\geq 1/2$.

Sharpness, Extremality, and Endpoint Analysis

The proof establishes both sharpness (equality for subcubes) and the failure of dimension-free bounds for exponents $\beta < 1/2$. Key to this is the collapse of the method for $\beta < 1/2$ (addressed via Hamming ball counterexamples).

The essential verification demands a mix of analytical computation in the regime near criticality (where previous work only established results for $\beta$ slightly above $1/2$) and interval arithmetic certified via computer-assisted proofs. The approach especially refines the analysis in regimes where the Bellman function transitions between its polynomial and Gaussian regimes. Six case subdivisions (Cases $J$, $Q$, $LJQ$, $LJ$, $QJQ$, $QJ$) comprehensively partition the space, ensuring all configurations obtain the optimal bound.

Applications and Consequences

Resolution of the Kahn-Park Conjecture

A direct consequence is the resolution of Kahn and Park’s cube-partition conjecture: for any partition $(A,B,W)$ of the hypercube with $|A|=1/2$, the edge boundary and residue size satisfy

$|\nabla(A,B)| + \sqrt{n}|W| \geq 1/2,$

which is optimal and achieved when $A$ is a half-cube and $W = \emptyset$.

Optimal $L^1$ Poincaré Inequality for Boolean Functions

The paper proves for all Boolean functions $f:\{0,1\}^n \to \{0,1\}$,

$\|\nabla f\|_1 \geq \|f - \mathbf{E}f\|_1,$

with equality on indicator functions of half-cubes. The bound improves upon prior estimates, surpassing the classical $L^1$ Poincaré constants known for general real-valued functions due to the additional structure for Boolean functions.

Information Theory: Connections to the Hellinger and Courtade-Kumar Conjectures

The sharp isoperimetric bound at $\beta=\frac{1}{2}$ provides, in the balanced case $|A|=1/2$, an explicit verification of the sensitivity inequality

$\mathbf{E}(\sqrt{s_f}) \geq 1$

for Boolean functions, where $s_f(x)$ denotes the sensitivity. This matches predictions from the Hellinger conjecture in information theory, specifically in the low-noise regime for Boolean channels, thereby aligning structural combinatorial inequalities with information-theoretic optimality criteria.

Computational Verification

The proof depends in crucial portions on certified numerical computation, leveraging rigorous interval arithmetic and recursive partitioning (via the FLINT/Arb library) to bound functional expressions arising in the Bellman function methodology. This highlights the increasing necessity and viability of computer-assisted proofs (especially in discrete geometric analysis), which now play a central role in verifying sharp threshold phenomena in high-dimensional combinatorics.

Implications and Outlook

On the theoretical side, this result closes a previously missing case in the hierarchy of isoperimetric inequalities on the discrete cube, mapping the landscape of exponents at and above the critical value $\beta=1/2$. It underscores the power of Bellman-function-based approaches in discrete settings and sets a new standard for sharp constants in Poincaré and isoperimetric inequalities for Boolean functions. Practically, the precise identification of minimal boundary-expectation underpins analysis of Boolean threshold phenomena, influences functional inequalities in random graph theory, and informs quantitative estimates in distributed computation and complexity.

The connection to information theory, particularly noisy channel analysis, further cements the relevance of isoperimetry in combinatorics to fundamental limits in communication and cryptography. The endpoint analysis at $\beta=\frac{1}{2}$ is also suggestive for future exploration in the extremal theory of functional inequalities for non-Boolean or vector-valued maps and for the search for corresponding results in the quantum and Banach-space-valued settings.

On the computational side, the reliance on certified numerics demonstrates evolving methodologies at the interface of discrete mathematics, analysis, and validated computation—pointing to future research that relies on or improves such machine-assistance, potentially leading to automated conjecture checking and further sharp threshold discoveries.

Conclusion

This work rigorously establishes the endpoint sharp isoperimetric inequality at the critical exponent $\beta=1/2$ for the Hamming cube, resolving conjectures in combinatorics and confirming predictions from information theory about Boolean function sensitivity. The analysis unifies advanced Bellman function constructions, detailed case analysis, and computer-assisted proof techniques. The resulting inequalities have direct implications for extremal combinatorics, Boolean analysis, and discrete probability, with promising avenues for further theoretical and computational investigation.