## 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.

### MSC:

 51M05 Euclidean geometries (general) and generalizations 53D12 Lagrangian submanifolds; Maslov index 51-03 History of geometry

### Citations:

Zbl 1196.57021; Zbl 1041.53049
Full Text:

### References:

 [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)
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