The rectangular peg problem. (English) Zbl 1472.51010

Greene, Joshua Evan; Lobb, Andrew

Ann. Math. (2) 194, No. 2, 509-517 (2021).

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.

Reviewer: Victor V. Pambuccian (Glendale)

MSC:

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

Keywords:

Jordan curve; aspect ratio; inscribed rectangles; symplectic geometry; Lagrangian surface

Citations:

Zbl 1196.57021; Zbl 1041.53049

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