Hostname: page-component-76d6cb85b7-xh428 Total loading time: 0 Render date: 2026-07-14T22:19:56.712Z Has data issue: false hasContentIssue false
  • English
  • Français

Polygons of unit area with vertices in sets of infinite planar measure

Part of: Classical measure theory

Published online by Cambridge University Press:  01 December 2025

Vjekoslav Kovač Open the ORCID record for Vjekoslav Kovač [Opens in a new window]  and
Bruno Predojević
Vjekoslav Kovač
Affiliation:
Faculty of Science, University of Zagreb , Croatia e-mail: vjekovac@math.hr
Bruno Predojević*
Affiliation:
Faculty of Science, University of Zagreb , Croatia e-mail: vjekovac@math.hr
*
e-mail: bruno.predojevic@math.hr
Get access Rights & Permissions [Opens in a new window]

Abstract

Paul Erdős and R. Daniel Mauldin asked a series of questions on certain types of polygons of area $1$, the vertices of which can be found in every planar set of infinite Lebesgue measure. We address two of these questions, one on cyclic quadrilaterals and the other on convex polygons with congruent sides, with, respectively, positive and negative answers.

Keywords

Geometric measure theorypoint configurationpolygonarea

MSC classification

Primary: 28A75: Length, area, volume, other geometric measure theory

Information

Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 16
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This work was supported in part by the Croatian Science Foundation under the project HRZZ-IP-2022-10-5116 (FANAP).

References

Adhikari, S. D. and Chen, Y.-G., On monochromatic configurations for finite colorings . Discrete Math. 333(2014), 106–109. https://doi.org/10.1016/j.disc.2014.06.015 CrossRefGoogle Scholar
Adhikari, S. D. and Rath, P., Remarks on monochromatic configurations for finite colorings of the plane . Note Mat. 32(2012), no. 2, 83–88. https://doi.org/10.1109/mcs.2012.2185888 Google Scholar
Bloom, T. F., Erdős problem #353. Accessed: December 16, 2024. https://www.erdosproblems.com/353 Google Scholar
Croft, H. T., Falconer, K. J., and Guy, R. K., Unsolved problems in geometry, Problem Books in Mathematics, Springer-Verlag, New York, NY, 1991. https://doi.org/10.1007/978-1-4612-0963-8 CrossRefGoogle Scholar
Dumitrescu, A. and Jiang, M., Monochromatic simplices of any volume . Discrete Math. 310(2010), no. 4, 956–960. https://doi.org/10.1016/j.disc.2009.09.026 CrossRefGoogle Scholar
Erdős, P., Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space . Real Anal. Exchange 4(1978/1979), no. 2, 113–138,10.2307/44151159CrossRefGoogle Scholar
Erdős, P., Some combinatorial, geometric and set theoretic problems in measure theory . In: Kölzow, D., and Maharam-Stone, D. (eds.), Measure theory, Oberwolfach 1983: proceedings of the conference held at Oberwolfach, June 26–July 2, 1983, volume 1089 of Lecture Notes in Mathematics, Springer, Berlin, 1984, pp. 321–327. https://doi.org/10.1007/BFb0072626 CrossRefGoogle Scholar
Folland, G. B., Real analysis: Modern techniques and their applications. 2nd ed., Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, NY, 1999. A Wiley-Interscience Publication.Google Scholar
Graham, R. L., On partitions of ${E}^n$ . J. Combin. Theory Ser. A. 28(1980), no. 1, 89–97. https://doi.org/10.1016/0097-3165(80)90061-8 CrossRefGoogle Scholar
Koizumi, J., Isosceles trapezoids of unit area with vertices in sets of infinite planar measure . Proc. Amer. Math. Soc. 153(2025), no. 11, 4753–4758. https://doi.org/10.1090/proc/17322 Google Scholar
Kovač, V., Coloring and density theorems for configurations of a given. Preprint, 2024. arXiv:2309.09973.Google Scholar
Stein, E. M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces, volume 32 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1971.Google Scholar
Steinhaus, H., Sur les distances des points dans les ensembles de mesure positive . Fund. Math. 1(1920), 93–104. https://doi.org/10.4064/fm-1-1-93-104 CrossRefGoogle Scholar