SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM

Jai Aslam, Shu-Jian Chen, Florian Frick, Sam Saloff-Coste, Linus Setiabrata, Hugh Thomas

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 $3$ -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 $d$ -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 inscribability results for the class of all continuous curves.

Locations

• Forum of Mathematics Sigma - View - PDF
• arXiv (Cornell University) - View - PDF