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. $C_{1a}$, $C_{1b}$).

IMPORTANT NOTE: while submissions to this site are reviewed to meet minimal standards of plausibility and replicability, they are not certified by this site for correctness, and may be subject to future revision, for instance due to errors in the associated preprint or paper. Thus, readers should exercise their own judgement when assessing the validity of the bounds reported on this site, particularly if their source is not yet published by a peer-reviewed journal. Bounds for which the level of available verification is currently at minimal levels will be marked with an asterisk in the table below.

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
3c	4-slope Kakeya-type sum-difference constant	1.67471	1.75
4a	Cap set constant	2.2202	2.756
4b	Furstenberg–Sárközy square-difference constant	0.733412	1
5a	Sidon set size constant	0	0.97633
5b	Sidon set density inside (4,5) sets	0.5294	0.5714
6	Union-closed sets conjecture constant	0.38271	0.5
7a	Irrationality measure of $\pi$	2	7.103205334137
7b	Irrationality measure of $\Gamma(1/4)$	2	$10^{143}$
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
10c	Spencer discrepancy constant (“six standard deviations suffice”)	1.414214	3.674235 (3.65*)
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	0.5
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
15a	Matrix multiplication exponent	2	2.371339
15b	Dual matrix multiplication exponent	>0.321334	1
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 (10.02*)
22b	Tight alternating knot constant	0.017	7.31
23a	Smallest unsolved instance of the Hadamard conjecture	668	$\infty$
23b	Minimal condition number decay for sign matrices	$\frac{17}{92}$	1
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)	$2$	$\infty$
27a	Chromatic number of the plane	5	7
27b	Maximum Chromatic Number of Biplanar Graphs	9	12
28	Smallest dimension in which Borsuk’s conjecture fails	4	64
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 (0.79970*)	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	$\frac{5}{4}$
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...
40b	Asymptotic Dobrowolski constant for Lehmer’s problem	$9/4$	$\infty$
41	Moving sofa constant	2.2195	2.37 (2.2195*)
42	Turan's pure power sum constant	0.5	0.69368
43	Gilbert-Pollak conjecture (Steiner ratio)	0.8559	0.86602540378
44	Maximal number of relevant variables in degree-$d$ Boolean functions	1.5	4.394
45	Density of odd integers that are the sum of a prime and a power of two	0.107648	0.490341088858244
46	Fourier restriction constant for the 2-sphere	3	$\frac{22}{7}\approx 3.142857$
47	Centered Hardy-Littlewood maximal constant in dimension $2$	$\frac{11+\sqrt{61}}{12}\approx 1.5675208$	9
48	Convex sub-Gaussian comparison constant	1	$<\infty$
49	Erdős–Szemerédi $3$-sunflower-free capacity	>1.551 ($\geq 1.554*$)	$\frac{3}{2^{2/3}} \approx 1.88988$
50	Approximation ratio for quantum Max Cut	0.611	$<1$ (0.5 for product states)
51	Erdős maximum term problem	0.5850788	$\frac{2}{\pi}\approx 0.63662$
52	Satisfiability threshold for random 3-SAT	3.52	4.490
53	Davenport constant for $C_n^3$	3	20369
54	Beurling–Ahlfors transform constant	1	1.575
55	Coefficient of the acyclic chromatic index	1	3.142
56	$\mathrm{GL}_2$ Ramanujan conjecture exponent	0	$\tfrac{7}{64}=0.109375$
57a	Bloch’s constant	$\frac{\sqrt{3}}{4}+2\times 10^{-4}$	$\dfrac{1}{\sqrt{1+\sqrt{3}}},\dfrac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\approx 0.4719$
57b	Landau's constant	$\frac{1}{2}+10^{-335}$	$\dfrac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\approx 0.5433$
57c	Univalent Bloch constant	0.5708858	1
58	Zaremba’s conjecture constant	5	$\infty$
59	Bohr radius for the bidisc	0.3006	0.3177
60	Favard-length decay exponent	$\frac{1}{6}$	1
61	Selberg congruence spectral-gap constant	0	$\frac{7}{64}$
62a	Lindelof (pointwise growth) exponent for the Riemann zeta function	0	$\frac{13}{84}$
62b	Burgess-quality subconvexity exponent for Dirichlet $L$-functions	0	$\frac{3}{16}$
63	Dirichlet divisor problem exponent	$\frac{1}{4}$	$\frac{131}{416}$
64	Gauss circle problem exponent	0	$\frac{131}{208}$
65	Linnik's constant	1	5
66	Elliott-Halberstam level-of-distribution exponent	$\frac{1}{2}$	1
67	Brennan's conjecture exponent	3.422	4
68	Korenblum's constant	0.28185	0.6778994
69	Sendov radius constant	1	2
70	Reverse Brunn-Minkowski constant	1	$<\infty$
71	Fourier Entropy-Influence constant	6.278	$\infty$
72	Polya-Vinogradov best constant (squarefree asymptotic)	0	$\frac{1}{4\pi}\approx 0.07958$
73	Flatness constant in dimension 3	$2+\sqrt{2}$	$<3.972$
74	10-point multi-point Seshadri constant on $\mathbb{P}^2$	$\frac{117}{370}$	$\frac{1}{\sqrt{10}}$
75	Metric TSP subtour-LP integrality-gap constant	$\frac{4}{3}$	$\frac{3}{2} - 2.18 \cdot 10^{-34}$
76	Asymptotic line-count constant for smooth degree-$d$ surfaces in $\mathbb{P}^3$ in characteristic $0$	3	11
77	3D critical Bochner–Riesz exponent	3	$\frac{13}{4}$
78	Conway thrackle constant	1	1.393
79	Asymptotic essential-dimension ratio of the symmetric groups	$\frac{1}{2}$	1

Recent progress

• 51 improved lower bound: $C_{51} \geq 0.5850724$ by Y. He and Q. Tang, 12 Feb 2026.
• 11b solved: $C_{11b} = 0.5$ by P. Durcik, P. Ivanisvili, J. Roos, X. Xie, 24 Feb 2026.
• 3c improved lower bound: $C_{3c} \geq 1.67471$ by T. Astor (paper coming soon).
• 5b improved upper and lower bounds: $\frac{9}{17} \leq C_{5b} \leq \frac{4}{7}$ by J. Ma and Q. Tang, 26 Feb 2026.
• 51 improved lower bound: $C_{51} \geq 0.5850788$ by N. Sothanaphan, 1 Mar 2026.
• 22 improved upper bound (unverified): $C_{22} \leq 10.02*$ by A. Klotz, 2 Mar 2026.
• 31 improved lower bound (unverified): $C_{31} \geq 0.79970*$ by Archivara, 4 Mar 2026.

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 = {http://github.com/teorth/optimizationproblems}
}

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