Inscribed Square Problem (Toeplitz)
Does every simple closed curve in the plane contain four points that form a square? The smooth and piecewise-linear cases have major partial results; the fully general Jordan curve remains open.
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- Validated increment
Smooth curves settled (Greene–Lobb)
Greene and Lobb proved that every smooth Jordan curve inscribes a square, via symplectic geometry. The remaining frontier is less regular curves: continuous or merely rectifiable Jordan curves.
Previous best
Squares for curves with extra smoothness / special classes
New certified
Squares for every C^∞ (in fact C^1) Jordan curve
Joshua Greene & Andrew Lobb
Also open