SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM
Forum of Mathematics, Sigma, Tome 8 (2020)
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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.
@article{10_1017_fms_2019_51,
author = {JAI ASLAM and SHUJIAN CHEN and FLORIAN FRICK and SAM SALOFF-COSTE and LINUS SETIABRATA and HUGH THOMAS},
title = {SPLITTING {LOOPS} {AND} {NECKLACES:} {VARIANTS} {OF} {THE} {SQUARE} {PEG} {PROBLEM}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {8},
year = {2020},
doi = {10.1017/fms.2019.51},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2019.51/}
}
TY - JOUR AU - JAI ASLAM AU - SHUJIAN CHEN AU - FLORIAN FRICK AU - SAM SALOFF-COSTE AU - LINUS SETIABRATA AU - HUGH THOMAS TI - SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM JO - Forum of Mathematics, Sigma PY - 2020 VL - 8 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.1017/fms.2019.51/ DO - 10.1017/fms.2019.51 LA - en ID - 10_1017_fms_2019_51 ER -
%0 Journal Article %A JAI ASLAM %A SHUJIAN CHEN %A FLORIAN FRICK %A SAM SALOFF-COSTE %A LINUS SETIABRATA %A HUGH THOMAS %T SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM %J Forum of Mathematics, Sigma %D 2020 %V 8 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.1017/fms.2019.51/ %R 10.1017/fms.2019.51 %G en %F 10_1017_fms_2019_51
JAI ASLAM; SHUJIAN CHEN; FLORIAN FRICK; SAM SALOFF-COSTE; LINUS SETIABRATA; HUGH THOMAS. SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM. Forum of Mathematics, Sigma, Tome 8 (2020). doi: 10.1017/fms.2019.51
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