Lipschitz normal embeddings in the space of matrices
A semi-algebraic subset in R n or C n is naturally equipped with two different metrics, the inner metric and the outer metric. Such a set (or its germ) is called Lipschitz normally embedded if the two metrics are bilipschitz equivalent. In this article we prove Lipschitz normal embeddedness of some...
Gespeichert in:
| Veröffentlicht in: | Mathematische Zeitschrift 2018-10, Vol.290 (1-2), p.485-507 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 507 |
|---|---|
| container_issue | 1-2 |
| container_start_page | 485 |
| container_title | Mathematische Zeitschrift |
| container_volume | 290 |
| creator | Kerner, Dmitry Pedersen, Helge Møller Ruas, Maria A. S. |
| description | A semi-algebraic subset in
R
n
or
C
n
is naturally equipped with two different metrics, the inner metric and the outer metric. Such a set (or its germ) is called
Lipschitz normally embedded
if the two metrics are bilipschitz equivalent. In this article we prove Lipschitz normal embeddedness of some algebraic subsets of the space of matrices. These include the space of rectangular/(skew-)symmetric/hermitian matrices of rank equal to a given number and their closures, and the upper triangular matrices with determinant 0. (In these cases we establish explicit bilipschitz constants.) We also make a short discussion about generalizing these results to determinantal varieties in real and complex spaces. |
| doi_str_mv | 10.1007/s00209-017-2027-4 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2095668285</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2095668285</sourcerecordid><originalsourceid>FETCH-LOGICAL-c316t-71a1ef4ed1cef2aa8ff6b3a4f924ca8260ac06e45f5f506e8bb0e5be6e1224733</originalsourceid><addsrcrecordid>eNp1kE9LAzEQxYMoWKsfwFvAc3SSzSbpUYpaYcGLnkM2O2m3dP-YbA_66U1ZwZPMYQbeezPMj5BbDvccQD8kAAErBlwzAUIzeUYWXBaCcSOKc7LIcslKo-UluUppD5BFLRdEVu2Y_K6dvmk_xM4dKHY1Nk3bbxNtezrtkKbReaRDoJ2bYusxXZOL4A4Jb377knw8P72vN6x6e3ldP1bMF1xNTHPHMUhsuMcgnDMhqLpwMqyE9M4IBc6DQlmGXHkwdQ1Y1qiQCyF1USzJ3bx3jMPnEdNk98Mx9vmkzc-WShlhyuzis8vHIaWIwY6x7Vz8shzsCY6d4dgMx57gWJkzYs6k7O23GP82_x_6AT-WZoQ</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2095668285</pqid></control><display><type>article</type><title>Lipschitz normal embeddings in the space of matrices</title><source>SpringerLink Journals</source><creator>Kerner, Dmitry ; Pedersen, Helge Møller ; Ruas, Maria A. S.</creator><creatorcontrib>Kerner, Dmitry ; Pedersen, Helge Møller ; Ruas, Maria A. S.</creatorcontrib><description>A semi-algebraic subset in
R
n
or
C
n
is naturally equipped with two different metrics, the inner metric and the outer metric. Such a set (or its germ) is called
Lipschitz normally embedded
if the two metrics are bilipschitz equivalent. In this article we prove Lipschitz normal embeddedness of some algebraic subsets of the space of matrices. These include the space of rectangular/(skew-)symmetric/hermitian matrices of rank equal to a given number and their closures, and the upper triangular matrices with determinant 0. (In these cases we establish explicit bilipschitz constants.) We also make a short discussion about generalizing these results to determinantal varieties in real and complex spaces.</description><identifier>ISSN: 0025-5874</identifier><identifier>EISSN: 1432-1823</identifier><identifier>DOI: 10.1007/s00209-017-2027-4</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Algebra ; Closures ; Mathematical analysis ; Mathematics ; Mathematics and Statistics ; Matrix methods ; Set theory</subject><ispartof>Mathematische Zeitschrift, 2018-10, Vol.290 (1-2), p.485-507</ispartof><rights>Springer-Verlag GmbH Germany, part of Springer Nature 2018</rights><rights>Copyright Springer Science & Business Media 2018</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-71a1ef4ed1cef2aa8ff6b3a4f924ca8260ac06e45f5f506e8bb0e5be6e1224733</citedby><cites>FETCH-LOGICAL-c316t-71a1ef4ed1cef2aa8ff6b3a4f924ca8260ac06e45f5f506e8bb0e5be6e1224733</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00209-017-2027-4$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00209-017-2027-4$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Kerner, Dmitry</creatorcontrib><creatorcontrib>Pedersen, Helge Møller</creatorcontrib><creatorcontrib>Ruas, Maria A. S.</creatorcontrib><title>Lipschitz normal embeddings in the space of matrices</title><title>Mathematische Zeitschrift</title><addtitle>Math. Z</addtitle><description>A semi-algebraic subset in
R
n
or
C
n
is naturally equipped with two different metrics, the inner metric and the outer metric. Such a set (or its germ) is called
Lipschitz normally embedded
if the two metrics are bilipschitz equivalent. In this article we prove Lipschitz normal embeddedness of some algebraic subsets of the space of matrices. These include the space of rectangular/(skew-)symmetric/hermitian matrices of rank equal to a given number and their closures, and the upper triangular matrices with determinant 0. (In these cases we establish explicit bilipschitz constants.) We also make a short discussion about generalizing these results to determinantal varieties in real and complex spaces.</description><subject>Algebra</subject><subject>Closures</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Matrix methods</subject><subject>Set theory</subject><issn>0025-5874</issn><issn>1432-1823</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><recordid>eNp1kE9LAzEQxYMoWKsfwFvAc3SSzSbpUYpaYcGLnkM2O2m3dP-YbA_66U1ZwZPMYQbeezPMj5BbDvccQD8kAAErBlwzAUIzeUYWXBaCcSOKc7LIcslKo-UluUppD5BFLRdEVu2Y_K6dvmk_xM4dKHY1Nk3bbxNtezrtkKbReaRDoJ2bYusxXZOL4A4Jb377knw8P72vN6x6e3ldP1bMF1xNTHPHMUhsuMcgnDMhqLpwMqyE9M4IBc6DQlmGXHkwdQ1Y1qiQCyF1USzJ3bx3jMPnEdNk98Mx9vmkzc-WShlhyuzis8vHIaWIwY6x7Vz8shzsCY6d4dgMx57gWJkzYs6k7O23GP82_x_6AT-WZoQ</recordid><startdate>20181001</startdate><enddate>20181001</enddate><creator>Kerner, Dmitry</creator><creator>Pedersen, Helge Møller</creator><creator>Ruas, Maria A. S.</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20181001</creationdate><title>Lipschitz normal embeddings in the space of matrices</title><author>Kerner, Dmitry ; Pedersen, Helge Møller ; Ruas, Maria A. S.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-71a1ef4ed1cef2aa8ff6b3a4f924ca8260ac06e45f5f506e8bb0e5be6e1224733</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>Algebra</topic><topic>Closures</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Matrix methods</topic><topic>Set theory</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Kerner, Dmitry</creatorcontrib><creatorcontrib>Pedersen, Helge Møller</creatorcontrib><creatorcontrib>Ruas, Maria A. S.</creatorcontrib><collection>CrossRef</collection><jtitle>Mathematische Zeitschrift</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Kerner, Dmitry</au><au>Pedersen, Helge Møller</au><au>Ruas, Maria A. S.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Lipschitz normal embeddings in the space of matrices</atitle><jtitle>Mathematische Zeitschrift</jtitle><stitle>Math. Z</stitle><date>2018-10-01</date><risdate>2018</risdate><volume>290</volume><issue>1-2</issue><spage>485</spage><epage>507</epage><pages>485-507</pages><issn>0025-5874</issn><eissn>1432-1823</eissn><abstract>A semi-algebraic subset in
R
n
or
C
n
is naturally equipped with two different metrics, the inner metric and the outer metric. Such a set (or its germ) is called
Lipschitz normally embedded
if the two metrics are bilipschitz equivalent. In this article we prove Lipschitz normal embeddedness of some algebraic subsets of the space of matrices. These include the space of rectangular/(skew-)symmetric/hermitian matrices of rank equal to a given number and their closures, and the upper triangular matrices with determinant 0. (In these cases we establish explicit bilipschitz constants.) We also make a short discussion about generalizing these results to determinantal varieties in real and complex spaces.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00209-017-2027-4</doi><tpages>23</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0025-5874 |
| ispartof | Mathematische Zeitschrift, 2018-10, Vol.290 (1-2), p.485-507 |
| issn | 0025-5874 1432-1823 |
| language | eng |
| recordid | cdi_proquest_journals_2095668285 |
| source | SpringerLink Journals |
| subjects | Algebra Closures Mathematical analysis Mathematics Mathematics and Statistics Matrix methods Set theory |
| title | Lipschitz normal embeddings in the space of matrices |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-04T12%3A29%3A15IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Lipschitz%20normal%20embeddings%20in%20the%20space%20of%20matrices&rft.jtitle=Mathematische%20Zeitschrift&rft.au=Kerner,%20Dmitry&rft.date=2018-10-01&rft.volume=290&rft.issue=1-2&rft.spage=485&rft.epage=507&rft.pages=485-507&rft.issn=0025-5874&rft.eissn=1432-1823&rft_id=info:doi/10.1007/s00209-017-2027-4&rft_dat=%3Cproquest_cross%3E2095668285%3C/proquest_cross%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2095668285&rft_id=info:pmid/&rfr_iscdi=true |