Square pegs between two graphs
We show that there always exists an inscribed square in a Jordan curve given as the union of two graphs of functions of Lipschitz constant less than $1 + \sqrt{2}$. We are motivated by Tao's result that there exists such a square in the case of Lipschitz constant less than $1$. In the case of L...
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| creator | Greene, Joshua Evan Lobb, Andrew |
| description | We show that there always exists an inscribed square in a Jordan curve given
as the union of two graphs of functions of Lipschitz constant less than $1 +
\sqrt{2}$. We are motivated by Tao's result that there exists such a square in
the case of Lipschitz constant less than $1$. In the case of Lipschitz constant
$1$, we show that the Jordan curve inscribes rectangles of every similarity
class. Our approach involves analysing the change in the spectral invariants of
the Jordan Floer homology under perturbations of the Jordan curve. |
| doi_str_mv | 10.48550/arxiv.2407.07798 |
| format | Article |
| fullrecord | <record><control><sourceid>arxiv_GOX</sourceid><recordid>TN_cdi_arxiv_primary_2407_07798</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2407_07798</sourcerecordid><originalsourceid>FETCH-arxiv_primary_2407_077983</originalsourceid><addsrcrecordid>eNpjYJA0NNAzsTA1NdBPLKrILNMzMjEw1zMwN7e04GSQCy4sTSxKVShITS9WSEotKU9NzVMoKc9XSC9KLMgo5mFgTUvMKU7lhdLcDPJuriHOHrpgk-ILijJzE4sq40EmxoNNNCasAgChIyo_</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Square pegs between two graphs</title><source>arXiv.org</source><creator>Greene, Joshua Evan ; Lobb, Andrew</creator><creatorcontrib>Greene, Joshua Evan ; Lobb, Andrew</creatorcontrib><description>We show that there always exists an inscribed square in a Jordan curve given
as the union of two graphs of functions of Lipschitz constant less than $1 +
\sqrt{2}$. We are motivated by Tao's result that there exists such a square in
the case of Lipschitz constant less than $1$. In the case of Lipschitz constant
$1$, we show that the Jordan curve inscribes rectangles of every similarity
class. Our approach involves analysing the change in the spectral invariants of
the Jordan Floer homology under perturbations of the Jordan curve.</description><identifier>DOI: 10.48550/arxiv.2407.07798</identifier><language>eng</language><subject>Mathematics - Combinatorics ; Mathematics - Geometric Topology ; Mathematics - Metric Geometry ; Mathematics - Symplectic Geometry</subject><creationdate>2024-07</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,778,883</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2407.07798$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2407.07798$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Greene, Joshua Evan</creatorcontrib><creatorcontrib>Lobb, Andrew</creatorcontrib><title>Square pegs between two graphs</title><description>We show that there always exists an inscribed square in a Jordan curve given
as the union of two graphs of functions of Lipschitz constant less than $1 +
\sqrt{2}$. We are motivated by Tao's result that there exists such a square in
the case of Lipschitz constant less than $1$. In the case of Lipschitz constant
$1$, we show that the Jordan curve inscribes rectangles of every similarity
class. Our approach involves analysing the change in the spectral invariants of
the Jordan Floer homology under perturbations of the Jordan curve.</description><subject>Mathematics - Combinatorics</subject><subject>Mathematics - Geometric Topology</subject><subject>Mathematics - Metric Geometry</subject><subject>Mathematics - Symplectic Geometry</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNpjYJA0NNAzsTA1NdBPLKrILNMzMjEw1zMwN7e04GSQCy4sTSxKVShITS9WSEotKU9NzVMoKc9XSC9KLMgo5mFgTUvMKU7lhdLcDPJuriHOHrpgk-ILijJzE4sq40EmxoNNNCasAgChIyo_</recordid><startdate>20240710</startdate><enddate>20240710</enddate><creator>Greene, Joshua Evan</creator><creator>Lobb, Andrew</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20240710</creationdate><title>Square pegs between two graphs</title><author>Greene, Joshua Evan ; Lobb, Andrew</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-arxiv_primary_2407_077983</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Mathematics - Combinatorics</topic><topic>Mathematics - Geometric Topology</topic><topic>Mathematics - Metric Geometry</topic><topic>Mathematics - Symplectic Geometry</topic><toplevel>online_resources</toplevel><creatorcontrib>Greene, Joshua Evan</creatorcontrib><creatorcontrib>Lobb, Andrew</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Greene, Joshua Evan</au><au>Lobb, Andrew</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Square pegs between two graphs</atitle><date>2024-07-10</date><risdate>2024</risdate><abstract>We show that there always exists an inscribed square in a Jordan curve given
as the union of two graphs of functions of Lipschitz constant less than $1 +
\sqrt{2}$. We are motivated by Tao's result that there exists such a square in
the case of Lipschitz constant less than $1$. In the case of Lipschitz constant
$1$, we show that the Jordan curve inscribes rectangles of every similarity
class. Our approach involves analysing the change in the spectral invariants of
the Jordan Floer homology under perturbations of the Jordan curve.</abstract><doi>10.48550/arxiv.2407.07798</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Combinatorics Mathematics - Geometric Topology Mathematics - Metric Geometry Mathematics - Symplectic Geometry |
| title | Square pegs between two graphs |
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