The square peg problem, posed by Toeplitz in 1911, asks whether every Jordan curve in the Euclidean plane contains the
vertices of a square. It remains open in this generality
(Greene and Lobb 2021). Schnirelman (1944) proved that a square can be inscribed
in any closed convex curve.
For smooth Jordan curves, Greene and Lobb (2021) proved the stronger rectangular peg theorem,
which prescribes the similarity class of the inscribed rectangle. Choosing that similarity
class to be a square gives the square peg conclusion for
smooth curves. However, because the rectangular peg theorem assumes smoothness, it
does not settle the square peg problem for arbitrary Jordan
curves. By contrast, a square can be circumscribed
about anyJordan curve (Steinhaus 1999, p. 104).
Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Inscribing Polygons in Curves." §B2 in Unsolved
Problems in Geometry. New York: Springer-Verlag, pp. 51-52, 1991.Greene,
J. E. and Lobb, A. "The Rectangular Peg Problem." Ann. Math.194,
509-517, 2021. https://doi.org/10.4007/annals.2021.194.2.4.Schnirelman,
L. G. "On Certain Geometrical Properties of Closed Curves." Uspehi
Matem. Nauk10, 34-44, 1944.Steinhaus, H. Mathematical
Snapshots, 3rd ed. New York: Dover, pp. 104 and 302, 1999.Toeplitz,
O. "Ueber einige Aufgaben der Analysis situs." Verh. Schweiz. Naturforsch.
Ges.94, 197, 1911.
