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).

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
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
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

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}
}