Carbery's rectangle problem
Let $A \subset [0, 1]^2$ be open with measure $\alpha$. Cauchy–Schwarz shows that four points of $A$ determine an axis-parallel rectangle of area $\gg \alpha^2 (\log 1/\alpha)^{-1}$, and a diagonal stripe of width about $\alpha$ shows that one cannot hope for area larger than order $\alpha^2$.
Problem. Is there an absolute constant $c > 0$ such that $A$ always contains the four vertices of an axis-parallel rectangle of area at least $c\alpha^2$?
Source: A. Carbery, M. Christ and J. Wright, Multidimensional van der Corput and sublevel set estimates, J. Amer. Math. Soc. 12 (1999), Section 6. Listed as Problem 85 in B. Green, 100 open problems (version of January 2026).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- CA — Classical Analysis and ODEs
- Secondary
- CO
- Keywords
- axis-parallel rectangles, harmonic analysis, measure theory, sublevel sets
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@article{carbery1999,
author = {Carbery, Anthony and Christ, Michael and Wright, James},
title = {Multidimensional van der {C}orput and sublevel set estimates},
journal = {Journal of the American Mathematical Society},
volume = {12},
number = {4},
pages = {981--1015},
year = {1999},
doi = {10.1090/S0894-0347-99-00309-4}
}
@article{keleti2000,
author = {Keleti, Tam{\'a}s},
title = {Density and covering properties of intervals of $\mathbb{R}^n$},
journal = {Mathematika},
volume = {47},
number = {1-2},
pages = {229--242},
year = {2000},
doi = {10.1112/S0025579300015849}
}
@article{katz1999,
author = {Katz, Nets Hawk},
title = {On the self crossing six sided figure problem},
journal = {New York Journal of Mathematics},
volume = {5},
pages = {121--130},
year = {1999}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 85; version of January 2026, most recent update December 2025},
year = {2026}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00956, "Carbery's rectangle problem", clopen.solutions.