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. In particular, one should not blindly use the results on this site for any research-level publication, without first checking the cited sources.

Bounds for which the level of available verification is currently at minimal levels will be marked with an asterisk in the table below. (This status may be updated if the verification status of the bound changes in the future, for instance if the preprint establishing the bound is published in a peer-reviewed journal.)

Number	Description	Best lower bound	Best upper bound
1a	Sidon set autocorrelation constant	1.2802 (1.292*)	1.502862
1b	Erdős minimum overlap constant	0.379005	0.380868
2	Crouzeix constant	2	2
3a	Gyarmati-Hennecart-Ruzsa sum-difference constant	1.19102809 (1.19519192*)	1.33333
3b	Kakeya sums-differences constant	1.77898 (1.77898884*)	1.83333
3c	4-slope Kakeya-type sum-difference constant	1.67473389 (1.6747338950414058*)	1.75
3d	Single-set sum-difference exponent	2	2
3e	Unnormalized single-set sum-difference exponent	1.27155	1.33333
4a	Cap set constant	2.2203	2.756
4b	Furstenberg–Sárközy square-difference constant	0.758068	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	4.896
9	Shannon capacity of the 7-cycle	3.25883262	3.3177
10a	The real Grothendieck constant	$\frac{6\pi}{11}\approx 1.71360$	$\frac{\pi}{2\log(1+\sqrt{2})} - 10^{-4} \approx 1.78211$
10b	The complex Grothendieck constant	1.338	1.40491
10c	Spencer discrepancy constant (“six standard deviations suffice”)	1.767767	4.1 (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	2.371177
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	$\sqrt{2} \approx 1.4142$	3.7919936995
18	Marton's conjecture constant (PFR)	1	9
19	Berry–Esseen constant	0.4097321837	0.4690 (0.423*)
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 (0.1875*)
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
23c	Asymptotic counting exponent for partial Hadamard matrices	1	3
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	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 (0.79970*)	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	2.6273856	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	$9/4$	$\infty$
41a	Moving sofa constant	2.2195	2.37 (2.2195*)
41b	Ambidextrous moving sofa constant	1.64495521	2.2195
42	Turan's pure power sum constant	>0.5	0.69368 (0.6906538*)
43	Gilbert-Pollak conjecture (Steiner ratio)	0.8559 (0.860*)	0.86602540378
44	Maximal number of relevant variables in Boolean functions of degree $d$	1.5	4.394
45	Density of odd integers that are the sum of a prime and a power of two	0.107648	0.490180063290061
46	Fourier restriction constant for the 2-sphere	3	$\frac{22}{7}\approx 3.142857$
47	Centered Hardy-Littlewood maximal constant in dimension $2$	1.68550999	3.879
48	One-dimensional convex sub-Gaussian comparison constant	$\approx 5.33386$	$\approx 5.33386$
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	0.614	$<1$ (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	4
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	0.302825279492
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	0.3554	0.6778994
69	Sendov radius constant	1	2
70	Reverse Brunn-Minkowski constant	1	$<\infty$
71	Fourier Entropy-Influence constant	6.278 (6.521845710923046575*)	$\infty$
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 surfaces of degree $d$ 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
80	Ising perceptron capacity threshold	>0 (0.833*)	0.847 (0.833*)
81	Brun's constant	1.840503	2.288513
82	Essential minimum of the Zhang-Zagier height	0.24874	0.2536331090204145
83	Wirsing Constant	0.30366300	0.30366300
84a	Erdős unit distance exponent	1.014 (1.03583*)	$\frac{4}{3}\approx 1.3333$
84b	Sum-product exponent for the reals	$\frac{4}{3}+\frac{10}{4407}\approx 1.3356$	$<2$ (1.999281*)
85	Exponent for commutators close to the identity	1	4
86	Schur–Siegel–Smyth trace constant	1.80203	1.8216
87	Martinet's constant for totally real number fields	60.8395	857.567
88a	Bounded prime gap constant	2	186
88b	Exponent for bounded gaps between many primes	0	3.8075

Recent progress

• 10c improved lower bound: $C_{10c} \geq 5/\sqrt{8} \approx 1.767767$ by Y. H., submitted to this repository, 5 Sep 2026.
• 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.
• 10a improved lower bound: $C_{10a} \geq 1.67696 + 10^{-12}$ by Chris Jones and Giulio Malavolta, 31 Mar 2026.
• 48 solved: $C_{48} = c_\star^2 \approx 5.33386$ by Damek Davis and Sam Power, 3 Apr 2026.
• 1a, 1b improved upper bounds: $C_{1a} \leq 1.503871$ and $C_{1b} \leq 0.380868$ by YLTLYSTYLLGDHZSWZSHMELCZX2026, 21 Apr 2026.
• 3a, 3c improved lower bounds: $C_{3a} \geq 1.1740744$ and $C_{3c} \geq 1.67473389$ by S. Griego, 13 May 2026.
• 84a improved lower bound (unverified): $C_{84a} \geq 1.03583*$ by E. Naslund, 25 May 2026.
• 84b improved upper bound (unverified): $C_{84b} \leq 1.999281*$ by I. Althoefer, 28 May 2026.
• 1a improved lower bound (unverified): $C_{1a} \geq 1.292*$ by A. Piterbarg, J. Bajaj, D. Vincent, 3 Jun 2026.
• 8 improved upper bound: $C_{8} \leq 4.896$ by C. Bellotti, T. Trudgian, A. Yang, 23 Mar 2026.
• 3a improved lower bound (limit value): $C_{3a} \geq 1.187326127925948*$ by Numaro, 23 Jul 2026.
• 3a improved lower bound (limit value): $C_{3a} \geq 1.19102809*$ by L. Kleinwaks, 24 Jul 2026.
• 3d solved: $C_{3d} = 2$ by H. Lin and S. Li, 29 Jul 2026.
• 3b improved lower bound (unverified): $C_{3b} \geq 1.77898884*$ by Mosaic Intelligence, entropy certificate on a 13-point support.
• 10a improved upper and lower bounds: $\frac{6\pi}{11} \leq C_{10a} \leq \frac{\pi}{2\log(1+\sqrt{2})} - 10^{-4}$ by R. Saha, A. Li, A. Xue, S. Chaudhuri, A. Klivans, P. K. Kothari, R. Meka, 11 Aug 2026 — determines the tenths digit of $C_{10a}$ to be $7$.
• 2 solved: $C_2 = 2$ — Crouzeix's conjecture, by S. Jin, 27 Jul 2026; an independent proof by a different route followed in E. Lorist and F. L. Schwenninger, 4 Aug 2026.
• 3a improved lower bound (limit value): $C_{3a} \geq 1.19519192*$ by L. Kleinwaks, 14 Aug 2026.
• 15a improved upper bound: $C_{15a} \leq 2.371177$ by E. Dupont, M. Eisenberger, B. Kozlovskii, A. Mehrabian, F. J. R. Ruiz, A. See, R. Zhou, J. Alman, V. Vassilevska Williams, M. Balog, 17 Aug 2026.
• 59 improved upper bound: $C_{59}=K_2<0.302825279492$ by Shivam Patel, 26 Aug 2026.
• 43 improved lower bound (unverified): $C_{43} \geq 0.860*$ (exact $43/50$; certificate-layer result conditional on the lemma set of KHSHGW2026) by J. Savva, 1 Sep 2026.
• 88a improved upper bound: $C_{88a} \leq 186$ via $\mathrm{DHL}[40,2]$, by OpenAI, 30 Aug 2026, with a Lean 4 formalization conditional on three declared axioms.
• 9 improved lower bound: $C_{9} \ge 134753^{1/10} \approx 3.258020$ by N. Itty, C. D. Rosin, C. Carstensen, D. Reichman, 23 Jul 2026.
• 9 improved lower bound: $C_{9} \ge 3.258789153908\ldots$ by Y. Gao, 30 Jul 2026.
• 9 improved lower bound: $C_{9} \ge 3.258805369885\ldots$ by P. Buys, S. Polak, J. Zuiddam, 31 Jul 2026, with a Lean 4 formalization.
• 9 improved lower bound: $C_{9} \ge 3.25883262\ldots$ by R. Tandon, 31 Aug 2026.
• 4b attribution correction: the $205/12$ lower bound $C_{4b}\ge 0.733412$ is due to R. Beigel and W. Gasarch, 2008; the same exponent was later published independently by M. Lewko, 2015.
• 4b improved lower bound: $C_{4b} \geq 0.752796$ by D. Krachun, 2 Aug 2026 — the first bound past $3/4$.
• 4b improved lower bound: $C_{4b} \geq 0.753742$ by JD Jones, 23 Aug 2026, with a Lean 4 formalization registered as Palomar entry PALOMAR-2026-08-26-000004 (standard axioms only).
• 4b improved lower bound: $C_{4b} \geq 0.758067$ (exact $37903373/50000000$) by E. Naslund, 19 Sep 2026, with a Lean 4 formalization registered as Palomar entry PALOMAR-2026-09-19-000006 (standard axioms only).
• 4b improved lower bound: $C_{4b} \geq 0.75806770413$ (exact $75806770413/10^{11}$) by JD Jones, 21 Sep 2026: Naslund's construction under his criterion with one composite code's words, the widths and all nine moment powers reallocated; Lean 4 formalization registered as Palomar entry PALOMAR-2026-09-21-000004 (standard axioms only).
• 3c improved lower bound (unverified): $C_{3c} \geq 1.6747338950414058*$ by Y. Lin, entropy certificate on a 147-point support, 9 Sep 2026.
• 19 improved upper bound (unverified): $C_{19} \leq 0.4395*$ by H. Xiao and C. Li, 10 Sep 2026.
• 19 improved upper bound (unverified): $C_{19} \leq 0.423*$ by Y. Lin, 13 Sep 2026.
• 47 improved lower bound: $C_{47} \geq 1.68550999$ by Y. Lin, 19 Sep 2026, with a Lean 4 formalization registered on Palomar.
• 47 improved upper bound: $C_{47} \leq 3.879$ by Y. Lin, 20 Sep 2026, formalized in Lean 4.
• 45 presentation: the printed certificate is Griego's $0.490249407811155$, not Yoo's record $0.490180063290061$.
• 10c upper bound correction: $C_{10c}\le 4.1$ by Pesenti–Vladu v2 (14 Apr 2026), replacing the withdrawn $3\sqrt{3/2}$ constant in Theorem 4.5.

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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