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SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM

Part of: Real and complex geometry Classical differential geometry

Published online by Cambridge University Press:  20 January 2020

JAI ASLAM ,
SHUJIAN CHEN ,
FLORIAN FRICK ,
SAM SALOFF-COSTE ,
LINUS SETIABRATA  and
HUGH THOMAS
JAI ASLAM
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC27695, USA; jkaslam@ncsu.edu
SHUJIAN CHEN
Affiliation:
Department of Mathematics, Brandeis University, Waltham, MA02453, USA; shujianchen@brandeis.edu
FLORIAN FRICK
Affiliation:
Department of Mathematics Sciences, Carnegie Mellon University, Pittsburgh, PA15213, USA; frick@cmu.edu
SAM SALOFF-COSTE
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY14853, USA; sps247@cornell.edu, ls823@cornell.edu
LINUS SETIABRATA
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY14853, USA; sps247@cornell.edu, ls823@cornell.edu
HUGH THOMAS
Affiliation:
Math. Dept., Université du Québec á Montréal, QuebecH2X 3Y7, Canada; thomas.hugh_r@uqam.ca

Abstract

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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.

MSC classification

Primary: 53A04: Curves in Euclidean space
Secondary: 51M04: Elementary problems in Euclidean geometries
Type
Analysis
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e5
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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