Splitting loops and necklaces: Variants of the square peg problem

Toeplitz conjectured that any simple planar loop inscribes a square. Here we prove variants of Toeplitz' square peg problem. We prove Hadwiger's 1971 conjecture that any simple loop in $3$-space inscribes a parallelogram. We show that any simple planar loop inscribes sufficiently many rectangles that their vertices are dense in the loop (independently due to Schwartz). If the loop is rectifiable, there is a rectangle that cuts the loop into four pieces that can be rearranged to form two loops of equal length. A rectifiable loop in $d$-space can be cut into $(r-1)(d+1)+1$ pieces that can be rearranged by translations to form $r$ 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 variants of Toeplitz' square peg problem for the class of all continuous curves.