I will describe how to use topological results from L^1 approximation theory to show that every spatial closed curve inscribes a parallelogram. This resolves a 1971 conjecture of Hadwiger. We will discuss related results for curves in other dimensions, in particular, planar curves. This is joint work with Jai Aslam, Shujian Chen, Sam Saloff-Coste, Linus Setiabrata, and Hugh Thomas.