Helly-Type Theorems for Line Transversals to Disjoint Unit Balls
Identifieur interne : 000289 ( Istex/Corpus ); précédent : 000288; suivant : 000290Helly-Type Theorems for Line Transversals to Disjoint Unit Balls
Auteurs : Otfried Cheong ; Xavier Goaoc ; Andreas Holmsen ; Sylvain PetitjeanSource :
- Discrete & Computational Geometry [ 0179-5376 ] ; 2008-03-01.
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Abstract
Abstract: We prove Helly-type theorems for line transversals to disjoint unit balls in ℝ d . In particular, we show that a family of n≥2d disjoint unit balls in ℝ d has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n≥4d−1 disjoint unit balls in ℝ d admits a line transversal if any subfamily of size 4d−1 admits a transversal.
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DOI: 10.1007/s00454-007-9022-1
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In particular, we show that a family of n≥2d disjoint unit balls in ℝ d has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. 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In particular, we show that a family of n≥2d disjoint unit balls in ℝ d has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. 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Balls</ArticleTitle><ArticleFirstPage>194</ArticleFirstPage><ArticleLastPage>212</ArticleLastPage><ArticleHistory><RegistrationDate><Year>2007</Year><Month>7</Month><Day>20</Day></RegistrationDate><Received><Year>2005</Year><Month>12</Month><Day>21</Day></Received><Revised><Year>2007</Year><Month>2</Month><Day>6</Day></Revised><OnlineDate><Year>2007</Year><Month>9</Month><Day>18</Day></OnlineDate></ArticleHistory><ArticleCopyright><CopyrightHolderName>Springer Science+Business Media, LLC</CopyrightHolderName><CopyrightYear>2007</CopyrightYear></ArticleCopyright><ArticleGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ArticleGrants></ArticleInfo><ArticleHeader><AuthorGroup><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>Otfried</GivenName><FamilyName>Cheong</FamilyName></AuthorName><Contact><Email>otfried@kaist.ac.kr</Email></Contact></Author><Author AffiliationIDS="Aff2" CorrespondingAffiliationID="Aff2"><AuthorName DisplayOrder="Western"><GivenName>Xavier</GivenName><FamilyName>Goaoc</FamilyName></AuthorName><Contact><Email>goaoc@loria.fr</Email></Contact></Author><Author AffiliationIDS="Aff3"><AuthorName DisplayOrder="Western"><GivenName>Andreas</GivenName><FamilyName>Holmsen</FamilyName></AuthorName><Contact><Email>andreash@mi.uib.no</Email></Contact></Author><Author AffiliationIDS="Aff4"><AuthorName DisplayOrder="Western"><GivenName>Sylvain</GivenName><FamilyName>Petitjean</FamilyName></AuthorName><Contact><Email>petitjea@loria.fr</Email></Contact></Author><Affiliation ID="Aff1"><OrgDivision>Division of Computer Science</OrgDivision><OrgName>KAIST</OrgName><OrgAddress><City>Daejeon</City><Country Code="KR">South Korea</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgName>LORIA–INRIA Lorraine</OrgName><OrgAddress><City>Nancy</City><Country Code="FR">France</Country></OrgAddress></Affiliation><Affiliation ID="Aff3"><OrgDivision>Department of Mathematics</OrgDivision><OrgName>University of Bergen</OrgName><OrgAddress><City>Bergen</City><Country Code="NO">Norway</Country></OrgAddress></Affiliation><Affiliation ID="Aff4"><OrgName>LORIA–CNRS</OrgName><OrgAddress><City>Nancy</City><Country Code="FR">France</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>
We prove Helly-type theorems for line transversals to disjoint unit balls in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>. In particular, we show that a family of <Emphasis Type="Italic">n</Emphasis>≥2<Emphasis Type="Italic">d</Emphasis> disjoint unit balls in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> has a line transversal if, for some ordering <Emphasis Type="Italic">≺</Emphasis> of the balls, any subfamily of 2<Emphasis Type="Italic">d</Emphasis> balls admits a line transversal consistent with <Emphasis Type="Italic">≺</Emphasis>. We also prove that a family of <Emphasis Type="Italic">n</Emphasis>≥4<Emphasis Type="Italic">d</Emphasis>−1 disjoint unit balls in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> admits a line transversal if any subfamily of size 4<Emphasis Type="Italic">d</Emphasis>−1 admits a transversal.
</Para></Abstract><KeywordGroup Language="En"><Heading>Keywords</Heading><Keyword>Geometric transversal theory</Keyword><Keyword>Helly-type theorem</Keyword><Keyword>Hadwiger-type theorem</Keyword><Keyword>Spheres</Keyword><Keyword>Balls</Keyword><Keyword>Line transversal</Keyword></KeywordGroup><ArticleNote Type="Misc"><SimplePara>Andreas Holmsen was supported by the Research Council of Norway, prosjektnummer 166618/V30. Otfried Cheong and Xavier Goaoc acknowledge support from the French-Korean Science and Technology Amicable Relationships program (STAR).</SimplePara></ArticleNote></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Helly-Type Theorems for Line Transversals to Disjoint Unit Balls</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>Helly-Type Theorems for Line Transversals to Disjoint Unit Balls</title></titleInfo><name type="personal"><namePart type="given">Otfried</namePart><namePart type="family">Cheong</namePart><affiliation>Division of Computer Science, KAIST, Daejeon, South Korea</affiliation><affiliation>E-mail: otfried@kaist.ac.kr</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal" displayLabel="corresp"><namePart type="given">Xavier</namePart><namePart type="family">Goaoc</namePart><affiliation>LORIA–INRIA Lorraine, Nancy, France</affiliation><affiliation>E-mail: goaoc@loria.fr</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Andreas</namePart><namePart type="family">Holmsen</namePart><affiliation>Department of Mathematics, University of Bergen, Bergen, Norway</affiliation><affiliation>E-mail: andreash@mi.uib.no</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Sylvain</namePart><namePart type="family">Petitjean</namePart><affiliation>LORIA–CNRS, Nancy, France</affiliation><affiliation>E-mail: petitjea@loria.fr</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Springer-Verlag</publisher><place><placeTerm type="text">New York</placeTerm></place><dateCreated encoding="w3cdtf">2005-12-21</dateCreated><dateIssued encoding="w3cdtf">2008-03-01</dateIssued><copyrightDate encoding="w3cdtf">2007</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: We prove Helly-type theorems for line transversals to disjoint unit balls in ℝ d . In particular, we show that a family of n≥2d disjoint unit balls in ℝ d has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n≥4d−1 disjoint unit balls in ℝ d admits a line transversal if any subfamily of size 4d−1 admits a transversal.</abstract><subject lang="en"><genre>Keywords</genre><topic>Geometric transversal theory</topic><topic>Helly-type theorem</topic><topic>Hadwiger-type theorem</topic><topic>Spheres</topic><topic>Balls</topic><topic>Line transversal</topic></subject><relatedItem type="host"><titleInfo><title>Discrete & Computational Geometry</title></titleInfo><titleInfo type="abbreviated"><title>Discrete Comput Geom</title></titleInfo><genre type="journal" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</genre><originInfo><publisher>Springer</publisher><dateIssued encoding="w3cdtf">2008-03-04</dateIssued><copyrightDate encoding="w3cdtf">2008</copyrightDate></originInfo><subject><genre>Mathematics</genre><topic>Computational Mathematics and Numerical Analysis</topic><topic>Combinatorics</topic></subject><identifier type="ISSN">0179-5376</identifier><identifier type="eISSN">1432-0444</identifier><identifier type="JournalID">454</identifier><identifier type="IssueArticleCount">29</identifier><identifier type="VolumeIssueCount">4</identifier><part><date>2008</date><detail type="volume"><number>39</number><caption>vol.</caption></detail><detail type="issue"><number>1-3</number><caption>no.</caption></detail><extent unit="pages"><start>194</start><end>212</end></extent></part><recordInfo><recordOrigin>Springer Science+Business Media, LLC, 2008</recordOrigin></recordInfo></relatedItem><identifier type="istex">0D4B2990174D778C492929582E432F7A738E9185</identifier><identifier type="ark">ark:/67375/VQC-M693GF0Q-2</identifier><identifier type="DOI">10.1007/s00454-007-9022-1</identifier><identifier type="ArticleID">9022</identifier><identifier type="ArticleID">s00454-007-9022-1</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer Science+Business Media, LLC, 2007</accessCondition><recordInfo><recordContentSource authority="ISTEX" authorityURI="https://loaded-corpus.data.istex.fr" valueURI="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-3XSW68JL-F">springer</recordContentSource><recordOrigin>Springer Science+Business Media, LLC, 2007</recordOrigin></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/ark:/67375/VQC-M693GF0Q-2/record.json</uri></json:item></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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