Splitting loops and necklaces: Variants of the square peg problem

Abstract

Toeplitz conjectured that any simple planar loop inscribes a square. Here we prove variants of Toeplitz's square peg problem. We prove Hadwiger's 1971 conjecture that any simple loop in-space inscribes a parallelogram. We show that any simple planar loop inscribes sufficiently many rectangles that their vertices are dense in the loop. If the loop is rectifiable, there is a rectangle that cuts the loop into four pieces which can be rearranged to form two loops of equal length. (The previous two results are independently due to Schwartz.) A rectifiable loop in-space can be cut into pieces that can be rearranged by translations to form loops of equal length. We relate our results to fair divisions of necklaces in the sense of Alon and to Tverberg-Type results. This provides a new approach and a common framework to obtain inscribability results for the class of all continuous curves.