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’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 th...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2020-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509419000513/type/journal_article |
| _version_ | 1811156169078603776 |
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| author | JAI ASLAM SHUJIAN CHEN FLORIAN FRICK SAM SALOFF-COSTE LINUS SETIABRATA HUGH THOMAS |
| author_facet | JAI ASLAM SHUJIAN CHEN FLORIAN FRICK SAM SALOFF-COSTE LINUS SETIABRATA HUGH THOMAS |
| author_sort | JAI ASLAM |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-10T04:47:02Z |
| format | Article |
| id | doaj.art-4433ce7df8e94150982457b1a37db65c |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:02Z |
| publishDate | 2020-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-4433ce7df8e94150982457b1a37db65c2023-03-09T12:34:47ZengCambridge University PressForum of Mathematics, Sigma2050-50942020-01-01810.1017/fms.2019.51SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEMJAI ASLAM0SHUJIAN CHEN1FLORIAN FRICK2SAM SALOFF-COSTE3LINUS SETIABRATA4HUGH THOMAS5Department of Mathematics, North Carolina State University, Raleigh, NC27695, USA;Department of Mathematics, Brandeis University, Waltham, MA02453, USA;Department of Mathematics Sciences, Carnegie Mellon University, Pittsburgh, PA15213, USA;Department of Mathematics, Cornell University, Ithaca, NY14853, USA; ,Department of Mathematics, Cornell University, Ithaca, NY14853, USA; ,Math. Dept., Université du Québec á Montréal, QuebecH2X 3Y7, Canada;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.https://www.cambridge.org/core/product/identifier/S2050509419000513/type/journal_article53A0451M04 |
| spellingShingle | 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 53A04 51M04 |
| title | SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM |
| title_full | SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM |
| title_fullStr | SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM |
| title_full_unstemmed | SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM |
| title_short | SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM |
| title_sort | splitting loops and necklaces variants of the square peg problem |
| topic | 53A04 51M04 |
| url | https://www.cambridge.org/core/product/identifier/S2050509419000513/type/journal_article |
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