Optimization Constants in Mathematics
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<br>Description<br>Best lower bound<br>Best upper bound
1a<br>Sidon set autocorrelation constant<br>1.2802<br>1.502862
1b<br>Erdős minimum overlap constant<br>0.379005<br>0.380868
Crouzeix constant<br>$1+\sqrt{2} \approx 2.4142$
3a<br>Gyamarti-Hennecart-Ruzsa sum-difference constant<br>1.1740744<br>1.33333
3b<br>Kakeya sums-differences constant<br>>1.77898<br>1.83333
3c<br>4-slope Kakeya-type sum-difference constant<br>1.67473389<br>1.75
4a<br>Cap set constant<br>2.2202<br>2.756
4b<br>Furstenberg–Sárközy square-difference constant<br>0.733412
5a<br>Sidon set size constant<br>0.97633
5b<br>Sidon set density inside (4,5) sets<br>0.5294<br>0.5714
Union-closed sets conjecture constant<br>0.38271<br>0.5
7a<br>Irrationality measure of $\pi$<br>7.103205334137
7b<br>Irrationality measure of $\Gamma(1/4)$<br>$10^{143}$
Classical zero-free region constant<br>0.755106<br>5.558691
Shannon capacity of the 7-cycle<br>3.2578<br>3.3177
10a<br>The real Grothendieck constant<br>$1.67696 + 10^{-26}$<br>1.782214
10b<br>The complex Grothendieck constant<br>1.338<br>1.40491
10c<br>Spencer discrepancy constant (“six standard deviations suffice”)<br>1.414214<br>3.674235 (3.65*)
11a<br>$L^1$ Poincaré constant on the Hamming cube<br>$\sqrt{\pi/2} \approx 1.2533$<br>$\pi/2 - 0.00013 \approx 1.5707$
11b<br>Critical exponent for isoperimetric inequality on the Hamming cube<br>0.5<br>0.5
12<br>The Beardwood–Halton–Hammersley constant<br>0.6277<br>0.90304
13a<br>Moser’s convex worm cover constant<br>0.232239<br>0.2617993878
13b<br>Lebesgue’s convex universal cover constant<br>0.832<br>0.8440935944
14<br>Smallest $n$ for which the value of $BB(n)$ is undecidable<br>432
15a<br>Matrix multiplication exponent<br>2.371339
15b<br>Dual matrix multiplication exponent<br>>0.321334
16<br>Brezis–Gallouet–Wainger remainder constant on the 2D torus<br>$\frac{\beta + \pi}{\pi} \approx 1.82283$<br>$\approx 2.15627$
17<br>Exponential growth constant of diagonal Ramsey numbers<br>$\sqrt{2} \approx 1.4142$<br>3.7992027396
18<br>Marton’s conjecture constant (PFR)
19<br>Berry–Esseen constant<br>0.4097321837<br>0.4690
20a<br>Thin shell conjecture constant
20b<br>Isotropic constant of a log-concave probability measure<br>$1/e$
20c<br>KLS constant for log-concave probability measures<br>$\sqrt{\pi/2} \approx 1.25331$<br>$\infty$
21<br>de Bruijn–Newman constant<br>0.2
22a<br>Tight knot constant<br>1.105<br>10.76 (10.02*)
22b<br>Tight alternating knot constant<br>0.017<br>7.31
23a<br>Smallest unsolved instance of the Hadamard conjecture<br>668<br>$\infty$
23b<br>Minimal condition number decay for sign matrices<br>$\frac{17}{92}$
23c<br>Asymptotic counting exponent for partial Hadamard matrices
24<br>Komlós discrepancy constant<br>$1+\sqrt{2}$<br>$\infty$
25<br>Mahler volume product constant<br>$\pi$
26a<br>Bohnenblust–Hille constant on the Boolean cube<br>$2$<br>$\infty$
26b<br>Multilinear Bohnenblust–Hille constant (real)<br>$2$<br>$\infty$
27a<br>Chromatic number of the plane
27b<br>Maximum Chromatic Number of Biplanar Graphs<br>12
28<br>Smallest dimension in which Borsuk’s conjecture fails<br>64
29<br>Kissing number in dimension $5$<br>40<br>44
30<br>Stanley–Wilf limit for the permutation pattern $1324$<br>10.27<br>13.5
31<br>Chvátal–Sankoff constant for a binary alphabet<br>0.792665992 (0.79970*)<br>0.826280
32<br>Constant term of one-shot channel simulation<br>$-\log_2 \log_2 e \approx -0.53$<br>$\sum_{k=1}^{\infty}2^{-k-1}k\log_{2}k-\log_{2}\log_{2}e \approx...