Optimization Constants in Mathematics

A curated collection of optimization constants $C$ in mathematics, often arising from solving a variational problem, or finding the best constant in a functional inequality. This repository is focused on recording the best known upper and lower bounds on constants that have an active literature, and encourages crowdsourced contributions and updates (see here for instructions on how to contribute).

• Here is an initial blog post introducing the project: A crowdsourced repository for optimization constants?, Terence Tao, 22 January 2026.

Table of Constants

We are arbitrarily numbering the constants as $C_{1}$, $C_{2}$, etc., mostly based on the order in which the constants were added to the repository. Constants that are in a family of similar constants will also be given letter suffixes (e.g. $C_{1a}$, $C_{1b}$).

Number	Description	Best lower bound	Best upper bound
1a	Sidon set autocorrelation constant	1.2802	1.5029
1b	Erdős minimum overlap constant	0.379005	0.380876
2	Crouzeix constant	2	$1+\sqrt{2} \approx 2.4142$
3a	Gyamarti-Hennecart-Ruzsa sum-difference constant	1.173077	1.33333
3b	Kakeya sums-differences constant	>1.77898	1.83333
4a	Cap set constant	2.2202	2.756
4b	Furstenberg–Sárközy square-difference constant	0.733412	1
5	Sidon set size constant	0	0.97633
6	Union-closed sets conjecture constant	0.38271	0.5
7	Irrationality measure of $\pi$	2	7.103205334137
8	Classical zero-free region constant	0.755106	5.558691
9	Shannon capacity of the 7-cycle	3.2578	3.3177
10a	The real Grothendieck constant	1.67696	1.782214
10b	The complex Grothendieck constant	1.338	1.40491
11a	$L^1$ Poincaré constant on the Hamming cube	$\sqrt{\pi/2} \approx 1.2533$	$\pi/2 - 0.00013 \approx 1.5707$
11b	Critical exponent for isoperimetric inequality on the Hamming cube	0.5	0.50057
12	The Beardwood–Halton–Hammersley constant	0.6277	0.90304
13a	Moser's convex worm cover constant	0.232239	0.2617993878
13b	Lebesgue's convex universal cover constant	0.832	0.8440935944
14	Smallest $n$ for which the value of $BB(n)$ is undecidable	6	432
15	Matrix multiplication exponent	2	2.371339
16	Brezis–Gallouet–Wainger remainder constant on the 2D torus	$\frac{\beta + \pi}{\pi} \approx 1.82283$	$\approx 2.15627$
17	Exponential growth constant of diagonal Ramsey numbers	$\sqrt{2} \approx 1.4142$	3.7992027396
18	Marton's conjecture constant (PFR)	1	9
19	Berry–Esseen constant	0.4097321837	0.4690
20a	Thin shell conjecture constant	2	$< \infty$
20b	Isotropic constant of a log-concave probability measure	$1/e$	$< \infty$
20c	KLS constant for log-concave probability measures	$\sqrt{\pi/2} \approx 1.25331$	$\infty$
21	de Bruijn–Newman constant	0	0.2
22a	Tight knot constant	1.105	10.76
22b	Tight alternating knot constant	0.017	7.31
23	Smallest unsolved instance of the Hadamard conjecture	668	$\infty$
24	Komlós discrepancy constant	$1+\sqrt{2}$	$\infty$
25	Mahler volume product constant	$\pi$	4
26a	Bohnenblust--Hille constant on the Boolean cube	$2$	$\infty$
26b	Multilinear Bohnenblust--Hille constant (real)	$2$	$\infty$
27	Chromatic number of the plane	5	7
28	Smallest dimension in which Borsuk’s conjecture fails	4	63
29	Kissing number in dimension $5$	40	44
30	Stanley–Wilf limit for the permutation pattern $1324$	10.27	13.5
31	Chvátal–Sankoff constant for a binary alphabet	0.792665992	0.826280
32	Constant term of one-shot channel simulation	$-\log_2 \log_2 e \approx -0.53$	$\sum_{k=1}^{\infty}2^{-k-1}k\log_{2}k-\log_{2}\log_{2}e \approx 0.76$
33	Ihara constant over $\mathbb{F}_2$	0.316999...	$\sqrt{2}-1 \approx 0.41421$
34	Falconer distance problem in $\mathbb{R}^2$	1	$5/4$
35	Gradient Descent Exponent	$\log_2(1+\sqrt{2}) \approx 1.271$	2
36	Sphere packing density in $\mathbb{R}^4$	$\pi^2/16 \approx 0.616850$	0.644421
37	The degree--sensitivity exponent	$\log_{3}(6) \approx 1.63093$	2
38	Square-lattice self-avoiding walk connective constant	2.625622	2.679193
39	Hadwiger covering / illumination number in $\mathbb{R}^3$	8	14
40a	Lehmer’s Mahler measure constant	1	1.176280...

Recent progress

• 11b solved: $C_{11b} = 0.5$ by P. Durcik, P. Ivanisvili, J. Roos, and X. Xie (paper coming soon).

Maintainers

This site is maintained by Damek Davis, Paata Ivanisvili and Terence Tao.

How to cite this repo

Use this BibTeX entry:

@misc{optimization-constants-repo,
  title = {Optimization Constants in Mathematics},
  author = {Davis, Damek and Ivanisvili, Paata and Tao, Terence and contributors},
  year = {2026},
  howpublished = {GitHub repository},
  url = {https://github.com/teorth/optimizationproblems}
}

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