Multilinear Bohnenblust–Hille constant (real)

For integers m,n≥ 1, let B_ℝ,m(n) be the smallest constant such that every m-linear form T:(ℓ_∞^n)^m → ℝ satisfies the (multilinear) Bohnenblust–Hille inequality (Σ_j_1,…,j_m=1^n bigl|T(e_j_1,…,e_j_m)bigr|^2m/m+1)^m+1/2m ≤ B_ℝ,m(n) ‖T‖, where ‖T‖:=sup_‖x^(1)‖_∞,…,‖x^(m)‖_∞ ≤…

Current state (tracked automatically)

Best known lower bound for constant 26b: 2 ([DMPSS2014], Proves the general lower bound B_ R,m≥ 2^1-1m for every m≥ 2).

All tracked facts and sources

Fact	Value	Source
Best lower bound with an explicit value (constant 26b)	2 [DMPSS2014] — Proves the general lower bound B_ R,m≥ 2^1-1m for every m≥ 2.	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-26b-multilinear-bohnenblust-hille-constant-real,
  title        = {Multilinear Bohnenblust–Hille constant (real)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-26b-multilinear-bohnenblust-hille-constant-real}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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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

For integers $m, n \geq 1$ , let $B_{R, m} (n)$ be the smallest constant such that every $m$ -linear form

$T : (ℓ_{\infty}^{n})^{m} \to R$

satisfies the (multilinear) Bohnenblust--Hille inequality

$(j_{1}, \dots, j_{m} = 1 \sum n T (e_{j_{1}}, \dots, e_{j_{m}})^{\frac{2 m}{m + 1}})^{\frac{m + 1}{2 m}} \leq B_{R, m} (n) ∥ T ∥,$

where

$∥ T ∥ := ∥ x^{(1)} ∥_{\infty}, \dots, ∥ x^{(m)} ∥_{\infty} \leq 1 sup T (x^{(1)}, \dots, x^{(m)}) .$

Define the optimal dimension-free (real) Bohnenblust--Hille constant of order $m$ by

$B_{R, m} := n \geq 1 sup B_{R, m} (n) .$

Finally, define $C_{26 b} := sup_{m \geq 1} B_{R, m} .$

Equivalently, $C_{26 b} < \infty$ if and only if the sequence $(B_{R, m})_{m \geq 1}$ is bounded.

Known upper bounds

Bound	Reference	Comments
$\infty$	Trivial	The best known general estimates on $B_{R, m}$ for each fixed $m$ are sublinear in $m$ ; for example $B_{R, m} < 1.3 m^{0.365}$ for $m \geq 14$ [CP2018].

Known lower bounds

Bound	Reference	Comments
$2$	[DMPSS2014]	Proves the general lower bound $B_{R, m} \geq 2^{1 - \frac{1}{m}}$ for every $m \geq 2$ . Taking $sup_{m}$ gives $C_{26 b} \geq 2$ . (For $m = 2$ this is sharp: $B_{R, 2} = 2$ , i.e. Littlewood's $4/3$ inequality.)

Additional comments and links

The exponent $\frac{2 m}{m + 1}$ in the multilinear Bohnenblust--Hille inequality is sharp. [CP2018]
Universality Conjecture [PT2016]. The optimal Bohnenblust--Hille constants should be bounded uniformly in $m$ ; in the real case, they conjecture the sharp values

$B_{R, m} = 2^{1 - \frac{1}{m}}$ for all $m$ ,

which would imply the exact value $C_{26 b} = 2$ .

See the survey [CP2018] for background, further references, and related polynomial/Hardy--Littlewood variants.

References

[BH1931] Bohnenblust, H. F.; Hille, E. On the absolute convergence of Dirichlet series. Ann. of Math. (2) 32 (1931), no. 3, 600--622.
[CP2018] Cavalcante, Wasthenny V.; Pellegrino, Daniel M. Bohnenblust--Hille inequalities: analytical and computational aspects. An. Acad. Bras. Ci\^enc. 91 (2019), suppl. 1, e20170398. doi:10.1590/0001-3765201720170398. (Epub 2018). Full text: https://www.scielo.br/j/aabc/a/TdCkK3xqRHNHgVx9g9VmSMp/?format=pdf&lang=en
[DMPSS2014] Diniz, D.; Mu\~noz-Fern\'andez, G. A.; Pellegrino, D.; Seoane-Sep\'ulveda, J. B. Lower bounds for the constants in the Bohnenblust--Hille inequality: the case of real scalars. Proc. Amer. Math. Soc. 142 (2014), no. 2, 575--580. https://arxiv.org/abs/1111.3253
[L1930] Littlewood, J. E. On bounded bilinear forms in an infinite number of variables. Quart. J. Math. 1 (1930), 164--174.
[PT2016] Pellegrino, Daniel M.; Teixeira, Eduardo. Sharp Bohnenblust--Hille constants for the mixed $(ℓ_{1}, ℓ_{2})$ -Littlewood inequality. (2016). https://arxiv.org/abs/1604.07595

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.