Helly-type theorems for intersections of sets starshaped via orthogonally convex paths
Let be a family of simply connected sets in the plane. If every countable subfamily of has an intersection that is starshaped via orthogonally convex paths, then itself has such an intersection. For the d -dimensional case, let be a family of compact sets in . If every finite subfamily of has an int...
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| Published in | Aequationes mathematicae Vol. 84; no. 3; pp. 207 - 217 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Basel
SP Birkhäuser Verlag Basel
01.12.2012
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0001-9054 1420-8903 |
| DOI | 10.1007/s00010-012-0151-0 |
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| Abstract | Let
be a family of simply connected sets in the plane. If every countable subfamily of
has an intersection that is starshaped via orthogonally convex paths, then
itself has such an intersection. For the
d
-dimensional case, let
be a family of compact sets in
. If every finite subfamily of
has an intersection that is starshaped via orthogonally convex paths, again
itself has such an intersection. |
|---|---|
| AbstractList | (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Let ... be a family of simply connected sets in the plane. If every countable subfamily of ... has an intersection that is starshaped via orthogonally convex paths, then ... itself has such an intersection. For the d-dimensional case, let ... be a family of compact sets in ... If every finite subfamily of ... has an intersection that is starshaped via orthogonally convex paths, again ... itself has such an intersection.[PUBLICATION ABSTRACT] Let $${\mathcal{K}}$$ be a family of simply connected sets in the plane. If every countable subfamily of $${\mathcal{K}}$$ has an intersection that is starshaped via orthogonally convex paths, then $${\mathcal{K}}$$ itself has such an intersection. For the d-dimensional case, let $${\mathcal{K}}$$ be a family of compact sets in $${\mathbb{R} delta }$$ . If every finite subfamily of $${\mathcal{K}}$$ has an intersection that is starshaped via orthogonally convex paths, again $${\mathcal{K}}$$ itself has such an intersection. Let be a family of simply connected sets in the plane. If every countable subfamily of has an intersection that is starshaped via orthogonally convex paths, then itself has such an intersection. For the d -dimensional case, let be a family of compact sets in . If every finite subfamily of has an intersection that is starshaped via orthogonally convex paths, again itself has such an intersection. |
| Author | Breen, Marilyn |
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| Keywords | Sets starshaped via orthogonally convex paths 52.A35 52.A30 |
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| References | Nadler, S.: Hyperspaces of Sets. New York (1978) BobylevN.A.The Helly theorem for star-shaped setsJ. Math. Sci.200110518191825187114610.1023/A:1011367730490 BreenM.Planar compact sets whose intersections are starshaped via orthogonally convex pathsAdv. Geom.2008816116924052341147.5200310.1515/ADVGEOM.2008.012 Valentine, F.A.: Convex Sets. New York (1964) BreenM.A Krasnosel’skii-type theorem for paths of bounded lengthArch. Math.1997686064142184710.1007/PL00000393 BreenM.Helly-type theorems for sets starshaped via staircase pathsBeiträge zur Algebra und Geometrie20084952753924680731180.52012 BreenM.Intersections of sets sharshaped via paths of bounded lengthMonatsh. Math.200313910510919879101033.5200510.1007/s00605-002-0512-1 KleeV.L.JrThe structure of semispacesMath. Scand.195645464809430070.39203 EckhoffJ.GruberP.M.WillsJ.M.Helly, Radon, and Carathéodory type theoremsHandbook of Convex Geometry, vol. A1993New YorkNorth Holland389448 LayS.R.Convex Sets and Their Applications1982New YorkWiley0492.52001 Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives, Convexity. In: Proceedings of the Symposium in Pure Mathematics, vol. 7, pp. 101–180. American Mathematical Society, Providence (1962) ApostolT.M.Mathematical Analysis1960ReadingAddison-Wesley BreenM.A Helly-type theorem for countable intersections of starshaped setsArch. Math.20058428228821341431080.5200610.1007/s00013-004-1120-1 M. Breen (151_CR4) 1997; 68 T.M. Apostol (151_CR1) 1960 M. Breen (151_CR5) 2008; 49 S.R. Lay (151_CR11) 1982 V.L. Klee Jr (151_CR10) 1956; 4 J. Eckhoff (151_CR9) 1993 M. Breen (151_CR6) 2003; 139 M. Breen (151_CR7) 2008; 8 151_CR8 151_CR13 151_CR12 N.A. Bobylev (151_CR2) 2001; 105 M. Breen (151_CR3) 2005; 84 |
| References_xml | – reference: BreenM.A Helly-type theorem for countable intersections of starshaped setsArch. Math.20058428228821341431080.5200610.1007/s00013-004-1120-1 – reference: ApostolT.M.Mathematical Analysis1960ReadingAddison-Wesley – reference: EckhoffJ.GruberP.M.WillsJ.M.Helly, Radon, and Carathéodory type theoremsHandbook of Convex Geometry, vol. A1993New YorkNorth Holland389448 – reference: KleeV.L.JrThe structure of semispacesMath. Scand.195645464809430070.39203 – reference: BreenM.A Krasnosel’skii-type theorem for paths of bounded lengthArch. Math.1997686064142184710.1007/PL00000393 – reference: BreenM.Intersections of sets sharshaped via paths of bounded lengthMonatsh. Math.200313910510919879101033.5200510.1007/s00605-002-0512-1 – reference: BreenM.Planar compact sets whose intersections are starshaped via orthogonally convex pathsAdv. Geom.2008816116924052341147.5200310.1515/ADVGEOM.2008.012 – reference: LayS.R.Convex Sets and Their Applications1982New YorkWiley0492.52001 – reference: Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives, Convexity. In: Proceedings of the Symposium in Pure Mathematics, vol. 7, pp. 101–180. American Mathematical Society, Providence (1962) – reference: BobylevN.A.The Helly theorem for star-shaped setsJ. Math. Sci.200110518191825187114610.1023/A:1011367730490 – reference: BreenM.Helly-type theorems for sets starshaped via staircase pathsBeiträge zur Algebra und Geometrie20084952753924680731180.52012 – reference: Nadler, S.: Hyperspaces of Sets. New York (1978) – reference: Valentine, F.A.: Convex Sets. New York (1964) – volume-title: Mathematical Analysis year: 1960 ident: 151_CR1 – volume: 68 start-page: 60 year: 1997 ident: 151_CR4 publication-title: Arch. Math. doi: 10.1007/PL00000393 – volume: 49 start-page: 527 year: 2008 ident: 151_CR5 publication-title: Beiträge zur Algebra und Geometrie – ident: 151_CR12 – ident: 151_CR13 – start-page: 389 volume-title: Handbook of Convex Geometry, vol. A year: 1993 ident: 151_CR9 doi: 10.1016/B978-0-444-89596-7.50017-1 – volume-title: Convex Sets and Their Applications year: 1982 ident: 151_CR11 – volume: 84 start-page: 282 year: 2005 ident: 151_CR3 publication-title: Arch. Math. doi: 10.1007/s00013-004-1120-1 – volume: 8 start-page: 161 year: 2008 ident: 151_CR7 publication-title: Adv. Geom. doi: 10.1515/ADVGEOM.2008.012 – volume: 4 start-page: 54 year: 1956 ident: 151_CR10 publication-title: Math. Scand. doi: 10.7146/math.scand.a-10455 – volume: 139 start-page: 105 year: 2003 ident: 151_CR6 publication-title: Monatsh. Math. doi: 10.1007/s00605-002-0512-1 – ident: 151_CR8 doi: 10.1090/pspum/007/0157289 – volume: 105 start-page: 1819 year: 2001 ident: 151_CR2 publication-title: J. Math. Sci. doi: 10.1023/A:1011367730490 |
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| Snippet | Let
be a family of simply connected sets in the plane. If every countable subfamily of
has an intersection that is starshaped via orthogonally convex paths,... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Let ... be a family of simply connected sets in the plane. If every countable... Let $${\mathcal{K}}$$ be a family of simply connected sets in the plane. If every countable subfamily of $${\mathcal{K}}$$ has an intersection that is... |
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| SubjectTerms | Analysis Combinatorics Intersections Mathematical analysis Mathematics Mathematics and Statistics Planes Theorems |
| Title | Helly-type theorems for intersections of sets starshaped via orthogonally convex paths |
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