The L^1 Poincaré constant on the Hamming cube

Current state (tracked automatically)

Best known bounds for constant 11a: 1.25331 ≤ C ≤ 0.00013. Upper bound ([IS2024], Provides an explicit integral expression for and evaluates it numerically (about 1.3 10^-4)); lower bound ([Pisier1986], [ILvHV2019], Comes from the sharp Gaussian L^1-Poincaré inequality (Pisier)).

All tracked facts and sources

Fact	Value	Source
Best lower bound with an explicit value (constant 11a)	1.25331 [Pisier1986], [ILvHV2019] — Comes from the sharp Gaussian L^1-Poincaré inequality (Pisier).	Tao's optimization constants
Best upper bound with an explicit value (constant 11a)	0.00013 [IS2024] — Provides an explicit integral expression for and evaluates it numerically (about 1.3 10^-4).	Tao's optimization constants

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Tao's optimization constants (checked 2026-11-12). How it works.

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Cite

@misc{cairn-constant-11a-the-l-1-poincare-constant-on-the-hamming-cube,
  title        = {The L^1 Poincaré constant on the Hamming cube},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-11a-the-l-1-poincare-constant-on-the-hamming-cube}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

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On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

Description of constant

$C_{11 a}$ is the smallest constant such that, for every $n \geq 1$ and every function $f$ :{-1,1}^n $\to R$

$E f (x) - E f (x) \leq C_{11 a} E ∣\nabla f ∣ (x),$

where $x = (x_{1}, \dots, x_{n})$ is uniform on {-1,1}^n and

$∣\nabla f ∣ (x) = (j = 1 \sum n ∣ D_{j} f (x) ∣^{2})^{1/2}, D_{j} f (x) = \frac{f ( x ) - f ( x ^{(j)} )}{2},$

with $x^{(j)} = (x_{1}, ..., x_{j - 1}, - x_{j}, x_{j + 1}, ..., x_{n}) .$

This is sometimes described as the (dimension-free) Cheeger constant appearing in the $L^{1}$ Poincaré inequality on the discrete cube.

Known upper bounds

Bound	Reference	Comments
$π /2 \approx 1.57080$	[BELP2008]	First proof (non-commutative/CAR algebra). Several later proofs recover the same constant.
$π /2 - δ$ for some $δ > 0$	[ILvHV2019]	First proof that $C_{11 a}$ is strictly smaller than $π /2$ .
$π /2 - δ$ with $δ \approx 0.00013$	[IS2024]	Provides an explicit integral expression for $δ$ and evaluates it numerically (about $1.3 \times 1 0^{- 4}$ ).

Known lower bounds

Bound	Reference	Comments
$1$	Trivial	For $n = 1$ , take $f (x) = x$ to get ratio $1$ .
$π /2 \approx 1.25331$	[Pisier1986], [ILvHV2019]	Comes from the sharp Gaussian $L^{1}$ -Poincaré inequality (Pisier).

Additional comments and links

It remains open whether $C_{11 a}$ equals $π /2$ .
arXiv:1811.05584, arXiv:2407.04835.

References

[BELP2008] Ben Efraim, L.; Lust-Piquard, F. Poincaré type inequalities on the discrete cube and in the CAR algebra. Probab. Theory Related Fields 141 (2008), no. 3–4, 569–602.
[ILvHV2019] Ivanisvili, P.; Li, D.; van Handel, R.; Volberg, A. Improving constant in end-point Poincaré inequality on Hamming cube. arXiv:1811.05584 (2018/2019).
[IS2024] Ivanisvili, P.; Stone, Y. Sharpening the gap between $L^{1}$ and $L^{2}$ norms. arXiv:2407.04835 (2024).
[Pisier1986] Pisier, G. Probabilistic methods in the geometry of Banach spaces. In: Probability and Analysis (Varenna, 1985), Lecture Notes in Math. 1206, Springer, Berlin (1986).

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.