The inscribed square problem
Let be any simple closed curve in the plane. Prove or disprove that always contains four points that are the vertices of a square.
History and references
Still open as of 2026.
Otto Toeplitz proposed the problem in 1911, and it is also known as the square peg problem. It has been proved for many classes of well-behaved curves. Arnold Emch handled piecewise analytic curves in 1916, and later work covers convex curves, piecewise smooth curves and locally monotone curves. The obvious strategy is to approximate an arbitrary curve by nice ones and pass to the limit, but the inscribed squares might shrink to a single point in the limit, and nobody has found a way to prevent that. For general continuous curves the question is still open.
References
Solutions 0
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