External Tangents and Closedness of Cone + Subspace
When X and Y are convex subsets of a topological vector space E, an external tangent of the ordered pair ( X, Y) is defined as an open ray T that issues from a point of X ∩ Y, is disjoint from X ∪ Y, and is such that X intersects each open halfspace containing T. It is shown that if E is a separable...
Gespeichert in:
| Veröffentlicht in: | Journal of mathematical analysis and applications 1994-12, Vol.188 (2), p.441-457 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 457 |
|---|---|
| container_issue | 2 |
| container_start_page | 441 |
| container_title | Journal of mathematical analysis and applications |
| container_volume | 188 |
| creator | Gritzmann, P. Klee, V. |
| description | When
X and
Y are convex subsets of a topological vector space
E, an
external tangent of the ordered pair (
X,
Y) is defined as an open ray
T that issues from a point of
X ∩
Y, is disjoint from
X ∪
Y, and is such that
X intersects each open halfspace containing
T. It is shown that if
E is a separable normed space,
C is a closed convex cone in
E, and
L is a line through the origin in
E, then the vector sum
C +
L = {
c + ℓ:
c ∈
C, ℓ ∈
L} is closed if and only if the pair (
C,
L) does not admit any external tangent. When
S is a subspace of finite dimension greater than 1, closedness of
C +
S is shown to be equivalent to the nonexistence of external tangents of a certain pair (
C′,
L,), where
L is a line through the origin and
C′ is a second closed convex cone constructed from (
C,
S). Questions about the closedness of sets of the form cone + subspace arise from Various optimization problems, from problems concerning the extension of positive linear functions, and from certain problems in matrix theory. |
| doi_str_mv | 10.1006/jmaa.1994.1438 |
| format | Article |
| fullrecord | <record><control><sourceid>elsevier_cross</sourceid><recordid>TN_cdi_crossref_primary_10_1006_jmaa_1994_1438</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><els_id>S0022247X84714387</els_id><sourcerecordid>S0022247X84714387</sourcerecordid><originalsourceid>FETCH-LOGICAL-c2278-790f31381723d02891e8a7ee5c9cc5481c9a6eb315768578d1dd3237b992afe63</originalsourceid><addsrcrecordid>eNp1jzFPwzAQhS0EEqWwMmdgQwk-O4ntEUWlIFVioEhs1tW-oFSpU9kFwb8nUREb0y3ve_c-xq6BF8B5fbfdIRZgTFlAKfUJmwE3dc41yFM241yIXJTq7ZxdpLTlHKBSMGNy8XWgGLDP1hjeKRxShsFnTT8k8oFSyoY2a4ZA2W328rFJe3R0yc5a7BNd_d45e31YrJvHfPW8fGruV7kTQulcGd5KkBqUkJ4LbYA0KqLKGeeqUoMzWNNGjkNqXSntwXsppNoYI7ClWs5Zcex1cUgpUmv3sdth_LbA7aRsJ2U7KdtJeQRujsAek8O-jRhcl_4oKRVoXY0xfYzROP6zo2iT6yg48l0kd7B-6P778APSqWfK</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>External Tangents and Closedness of Cone + Subspace</title><source>Elsevier ScienceDirect Journals</source><source>Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals</source><creator>Gritzmann, P. ; Klee, V.</creator><creatorcontrib>Gritzmann, P. ; Klee, V.</creatorcontrib><description>When
X and
Y are convex subsets of a topological vector space
E, an
external tangent of the ordered pair (
X,
Y) is defined as an open ray
T that issues from a point of
X ∩
Y, is disjoint from
X ∪
Y, and is such that
X intersects each open halfspace containing
T. It is shown that if
E is a separable normed space,
C is a closed convex cone in
E, and
L is a line through the origin in
E, then the vector sum
C +
L = {
c + ℓ:
c ∈
C, ℓ ∈
L} is closed if and only if the pair (
C,
L) does not admit any external tangent. When
S is a subspace of finite dimension greater than 1, closedness of
C +
S is shown to be equivalent to the nonexistence of external tangents of a certain pair (
C′,
L,), where
L is a line through the origin and
C′ is a second closed convex cone constructed from (
C,
S). Questions about the closedness of sets of the form cone + subspace arise from Various optimization problems, from problems concerning the extension of positive linear functions, and from certain problems in matrix theory.</description><identifier>ISSN: 0022-247X</identifier><identifier>EISSN: 1096-0813</identifier><identifier>DOI: 10.1006/jmaa.1994.1438</identifier><identifier>CODEN: JMANAK</identifier><language>eng</language><publisher>San Diego, CA: Elsevier Inc</publisher><subject>Convex and discrete geometry ; Exact sciences and technology ; General topology ; Geometry ; Mathematics ; Sciences and techniques of general use ; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds</subject><ispartof>Journal of mathematical analysis and applications, 1994-12, Vol.188 (2), p.441-457</ispartof><rights>1994 Academic Press</rights><rights>1995 INIST-CNRS</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/S0022247X84714387$$EHTML$$P50$$Gelsevier$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,3537,27901,27902,65306</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=3371885$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Gritzmann, P.</creatorcontrib><creatorcontrib>Klee, V.</creatorcontrib><title>External Tangents and Closedness of Cone + Subspace</title><title>Journal of mathematical analysis and applications</title><description>When
X and
Y are convex subsets of a topological vector space
E, an
external tangent of the ordered pair (
X,
Y) is defined as an open ray
T that issues from a point of
X ∩
Y, is disjoint from
X ∪
Y, and is such that
X intersects each open halfspace containing
T. It is shown that if
E is a separable normed space,
C is a closed convex cone in
E, and
L is a line through the origin in
E, then the vector sum
C +
L = {
c + ℓ:
c ∈
C, ℓ ∈
L} is closed if and only if the pair (
C,
L) does not admit any external tangent. When
S is a subspace of finite dimension greater than 1, closedness of
C +
S is shown to be equivalent to the nonexistence of external tangents of a certain pair (
C′,
L,), where
L is a line through the origin and
C′ is a second closed convex cone constructed from (
C,
S). Questions about the closedness of sets of the form cone + subspace arise from Various optimization problems, from problems concerning the extension of positive linear functions, and from certain problems in matrix theory.</description><subject>Convex and discrete geometry</subject><subject>Exact sciences and technology</subject><subject>General topology</subject><subject>Geometry</subject><subject>Mathematics</subject><subject>Sciences and techniques of general use</subject><subject>Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds</subject><issn>0022-247X</issn><issn>1096-0813</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1994</creationdate><recordtype>article</recordtype><recordid>eNp1jzFPwzAQhS0EEqWwMmdgQwk-O4ntEUWlIFVioEhs1tW-oFSpU9kFwb8nUREb0y3ve_c-xq6BF8B5fbfdIRZgTFlAKfUJmwE3dc41yFM241yIXJTq7ZxdpLTlHKBSMGNy8XWgGLDP1hjeKRxShsFnTT8k8oFSyoY2a4ZA2W328rFJe3R0yc5a7BNd_d45e31YrJvHfPW8fGruV7kTQulcGd5KkBqUkJ4LbYA0KqLKGeeqUoMzWNNGjkNqXSntwXsppNoYI7ClWs5Zcex1cUgpUmv3sdth_LbA7aRsJ2U7KdtJeQRujsAek8O-jRhcl_4oKRVoXY0xfYzROP6zo2iT6yg48l0kd7B-6P778APSqWfK</recordid><startdate>19941201</startdate><enddate>19941201</enddate><creator>Gritzmann, P.</creator><creator>Klee, V.</creator><general>Elsevier Inc</general><general>Elsevier</general><scope>6I.</scope><scope>AAFTH</scope><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>19941201</creationdate><title>External Tangents and Closedness of Cone + Subspace</title><author>Gritzmann, P. ; Klee, V.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c2278-790f31381723d02891e8a7ee5c9cc5481c9a6eb315768578d1dd3237b992afe63</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1994</creationdate><topic>Convex and discrete geometry</topic><topic>Exact sciences and technology</topic><topic>General topology</topic><topic>Geometry</topic><topic>Mathematics</topic><topic>Sciences and techniques of general use</topic><topic>Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Gritzmann, P.</creatorcontrib><creatorcontrib>Klee, V.</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>Pascal-Francis</collection><collection>CrossRef</collection><jtitle>Journal of mathematical analysis and applications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Gritzmann, P.</au><au>Klee, V.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>External Tangents and Closedness of Cone + Subspace</atitle><jtitle>Journal of mathematical analysis and applications</jtitle><date>1994-12-01</date><risdate>1994</risdate><volume>188</volume><issue>2</issue><spage>441</spage><epage>457</epage><pages>441-457</pages><issn>0022-247X</issn><eissn>1096-0813</eissn><coden>JMANAK</coden><abstract>When
X and
Y are convex subsets of a topological vector space
E, an
external tangent of the ordered pair (
X,
Y) is defined as an open ray
T that issues from a point of
X ∩
Y, is disjoint from
X ∪
Y, and is such that
X intersects each open halfspace containing
T. It is shown that if
E is a separable normed space,
C is a closed convex cone in
E, and
L is a line through the origin in
E, then the vector sum
C +
L = {
c + ℓ:
c ∈
C, ℓ ∈
L} is closed if and only if the pair (
C,
L) does not admit any external tangent. When
S is a subspace of finite dimension greater than 1, closedness of
C +
S is shown to be equivalent to the nonexistence of external tangents of a certain pair (
C′,
L,), where
L is a line through the origin and
C′ is a second closed convex cone constructed from (
C,
S). Questions about the closedness of sets of the form cone + subspace arise from Various optimization problems, from problems concerning the extension of positive linear functions, and from certain problems in matrix theory.</abstract><cop>San Diego, CA</cop><pub>Elsevier Inc</pub><doi>10.1006/jmaa.1994.1438</doi><tpages>17</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0022-247X |
| ispartof | Journal of mathematical analysis and applications, 1994-12, Vol.188 (2), p.441-457 |
| issn | 0022-247X 1096-0813 |
| language | eng |
| recordid | cdi_crossref_primary_10_1006_jmaa_1994_1438 |
| source | Elsevier ScienceDirect Journals; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | Convex and discrete geometry Exact sciences and technology General topology Geometry Mathematics Sciences and techniques of general use Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds |
| title | External Tangents and Closedness of Cone + Subspace |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-02-06T18%3A53%3A20IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-elsevier_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=External%20Tangents%20and%20Closedness%20of%20Cone%20+%20Subspace&rft.jtitle=Journal%20of%20mathematical%20analysis%20and%20applications&rft.au=Gritzmann,%20P.&rft.date=1994-12-01&rft.volume=188&rft.issue=2&rft.spage=441&rft.epage=457&rft.pages=441-457&rft.issn=0022-247X&rft.eissn=1096-0813&rft.coden=JMANAK&rft_id=info:doi/10.1006/jmaa.1994.1438&rft_dat=%3Celsevier_cross%3ES0022247X84714387%3C/elsevier_cross%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_id=info:pmid/&rft_els_id=S0022247X84714387&rfr_iscdi=true |