In 1911 Toeplitz conjectured that any simple continuous closed curve in the plane inscribes a square. A less famous variant of this problem is Hadwiger's 1971 conjecture that any simple closed continuous curve in 3-space inscribes a parallelogram. Both conjectures have been resolved under some smoothness condition on the curve. I will survey some of the known results and then report on recent progress on both conjectures. In particular we resolve Hadwiger's conjecture in full generality by relating it to partition results for real-valued functions.
This is joint work with Jai Aslam, Shujian Chen, Sam Saloff-Coste, Linus Setiabrata, and Hugh Thomas.