The rectangular peg problem (Q1983099)

Using a theorem of \textit{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 \textit{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.