The rectangular peg problem. (English) Zbl 1472.51010

Using a theorem of V. Shevchishin [Izv. Math. 73, No. 4, 797–859 (2009); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 73, No. 4, 153–224 (2009; Zbl 1196.57021)] and S. Yu. Nemirovski [Izv. Math. 66, No. 1, 151–164 (2002); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 66, No. 1, 153–166 (2002; Zbl 1041.53049)] – that the Klein bottle does not admit a smooth Lagrangian embedding in \({\mathbb C}^2\) –, a proposition proved in this paper regarding a Lagrangian smoothing, involving a \(4\)-manifold with a symplectic form, and the equivariant Darboux-Weinstein theorem, the authors prove the long-standing conjecture that for every smooth Jordan curve \(\gamma\) and rectangle \(R\) in the Euclidean plane, there exists a rectangle similar to \(R\) whose vertices lie on \(\gamma\).
The paper ends with an in-depth history of the problem, going back to Toeplitz in 1911.


51M05 Euclidean geometries (general) and generalizations
53D12 Lagrangian submanifolds; Maslov index
51-03 History of geometry
Full Text: DOI arXiv Link


[1] Batson, Joshua, Nonorientable slice genus can be arbitrarily large, Math. Res. Lett.. Mathematical Research Letters, 21, 423-436 (2014) · Zbl 1308.57004
[2] Dwivedi, Shubham; Herman, Jonathan; Jeffrey, Lisa C.; van den Hurk, Theo, Hamiltonian group actions and equivariant cohomology, SpringerBriefs in Mathematics, xi+132 pp. (2019) · Zbl 1447.53002
[3] Emch, Arnold, Some {P}roperties of {C}losed {C}onvex {C}urves in a {P}lane, Amer. J. Math.. American Journal of Mathematics, 35, 407-412 (1913) · JFM 44.0561.01
[4] Feller, P.; Golla, M., Non-orientable slice surfaces and inscribed rectangles (2020)
[5] Griffiths, H. B., The topology of square pegs in round holes, Proc. London Math. Soc. (3). Proceedings of the London Mathematical Society. Third Series, 62, 647-672 (1991) · Zbl 0696.55003
[6] Hugelmeyer, C., Every smooth {J}ordan curve has an inscribed rectangle with aspect ratio equal to \(\sqrt{3} (2018)\)
[7] Hugelmeyer, C., Inscribed rectangles in a smooth {J}ordan curve attain at least one third of all aspect ratios, Ann. of Math. (2). Annals of Mathematics. Second Series, 194, 497-508 (2021)
[8] Klee, Victor; Wagon, Stan, Old and new unsolved problems in plane geometry and number theory, The Dolciani Mathematical Expositions, 11, xvi+333 pp. (1991) · Zbl 0784.51002
[9] Mak, Cheuk Yu; Wu, Weiwei, Dehn twist exact sequences through {L}agrangian cobordism, Compos. Math.. Compositio Mathematica, 154, 2485-2533 (2018) · Zbl 07036911
[10] Matschke, Benjamin, A survey on the square peg problem, Notices Amer. Math. Soc.. Notices of the American Mathematical Society, 61, 346-352 (2014) · Zbl 1338.51017
[11] Meyerson, Mark D., Balancing acts, Topology Proc.. Topology Proceedings, 6, 59-75 (1981) · Zbl 0493.57003
[12] Nemirovski\u{\i}, S. Yu., The homology class of a {L}agrangian {K}lein bottle, Izv. Ross. Akad. Nauk Ser. Mat.. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 73, 37-48 (2009) · Zbl 1191.57019
[13] Pak, I., The discrete square peg problem (2008)
[14] Po{\'z}niak, M., Floer homology, {N}ovikov rings and clean intersections (1994) · Zbl 0948.57025
[15] \v{S}nirel\cprime man, L. G., On certain geometrical properties of closed curves, Uspehi Matem. Nauk. Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 10, 34-44 (1944)
[16] Schwartz, Richard Evan, A trichotomy for rectangles inscribed in {J}ordan loops, Geom. Dedicata. Geometriae Dedicata, 208, 177-196 (2020) · Zbl 1448.51008
[17] Shevchishin, V. V., Lagrangian embeddings of the {K}lein bottle and the combinatorial properties of mapping class groups, Izv. Ross. Akad. Nauk Ser. Mat.. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 73, 153-224 (2009) · Zbl 1196.57021
[18] Tao, Terence, An integration approach to the {T}oeplitz square peg problem, Forum Math. Sigma. Forum of Mathematics. Sigma, 5, 30-63 (2017) · Zbl 1422.52001
[19] Toeplitz, O., Ueber einige {A}ufgaben der {A}nalysis situs, Verhandlungen der {S}chweizerischen {N}aturforschenden {G}esellschaft,, 197 pp. (1911)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.