A Refined Energy Bound for Distinct Perpendicular Bisectors
Abstract Let $${\mathcal {P}}$$ be a set of n points in the Euclidean plane. We prove that, for any $$\varepsilon > 0$$, either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors determined by pairs of points in $${\mathcal {P}}$$...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Lund, Ben [verfasserIn] |
|---|
| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2020 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Nature Switzerland AG 2020 |
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| Übergeordnetes Werk: |
Enthalten in: Annals of combinatorics - Springer International Publishing, 1997, 24(2020), 2 vom: 14. Jan., Seite 225-235 |
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| Übergeordnetes Werk: |
volume:24 ; year:2020 ; number:2 ; day:14 ; month:01 ; pages:225-235 |
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| DOI / URN: |
10.1007/s00026-019-00478-z |
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OLC2061537642 |
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| 520 | |a Abstract Let $${\mathcal {P}}$$ be a set of n points in the Euclidean plane. We prove that, for any $$\varepsilon > 0$$, either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors determined by pairs of points in $${\mathcal {P}}$$ is $$\Omega (n^{52/35 - \varepsilon })$$, where the constant implied by the $$\Omega $$ notation depends on $$\varepsilon $$. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors is $$\Omega (n^2)$$. The proof relies bounding the size of a carefully selected subset of the quadruples $$(a,b,c,d) \in {\mathcal {P}}^4$$ such that the perpendicular bisector of a and b is the same as the perpendicular bisector of c and d. | ||
| 650 | 4 | |a Incidences | |
| 650 | 4 | |a Perpendicular bisectors | |
| 650 | 4 | |a Distinct distances | |
| 650 | 4 | |a Energy bound | |
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10.1007/s00026-019-00478-z doi (DE-627)OLC2061537642 (DE-He213)s00026-019-00478-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Lund, Ben verfasserin aut A Refined Energy Bound for Distinct Perpendicular Bisectors 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Nature Switzerland AG 2020 Abstract Let $${\mathcal {P}}$$ be a set of n points in the Euclidean plane. We prove that, for any $$\varepsilon > 0$$, either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors determined by pairs of points in $${\mathcal {P}}$$ is $$\Omega (n^{52/35 - \varepsilon })$$, where the constant implied by the $$\Omega $$ notation depends on $$\varepsilon $$. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors is $$\Omega (n^2)$$. The proof relies bounding the size of a carefully selected subset of the quadruples $$(a,b,c,d) \in {\mathcal {P}}^4$$ such that the perpendicular bisector of a and b is the same as the perpendicular bisector of c and d. Incidences Perpendicular bisectors Distinct distances Energy bound Enthalten in Annals of combinatorics Springer International Publishing, 1997 24(2020), 2 vom: 14. Jan., Seite 225-235 (DE-627)234146176 (DE-600)1394631-6 (DE-576)094078408 0218-0006 nnns volume:24 year:2020 number:2 day:14 month:01 pages:225-235 https://doi.org/10.1007/s00026-019-00478-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_267 AR 24 2020 2 14 01 225-235 |
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10.1007/s00026-019-00478-z doi (DE-627)OLC2061537642 (DE-He213)s00026-019-00478-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Lund, Ben verfasserin aut A Refined Energy Bound for Distinct Perpendicular Bisectors 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Nature Switzerland AG 2020 Abstract Let $${\mathcal {P}}$$ be a set of n points in the Euclidean plane. We prove that, for any $$\varepsilon > 0$$, either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors determined by pairs of points in $${\mathcal {P}}$$ is $$\Omega (n^{52/35 - \varepsilon })$$, where the constant implied by the $$\Omega $$ notation depends on $$\varepsilon $$. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors is $$\Omega (n^2)$$. The proof relies bounding the size of a carefully selected subset of the quadruples $$(a,b,c,d) \in {\mathcal {P}}^4$$ such that the perpendicular bisector of a and b is the same as the perpendicular bisector of c and d. Incidences Perpendicular bisectors Distinct distances Energy bound Enthalten in Annals of combinatorics Springer International Publishing, 1997 24(2020), 2 vom: 14. Jan., Seite 225-235 (DE-627)234146176 (DE-600)1394631-6 (DE-576)094078408 0218-0006 nnns volume:24 year:2020 number:2 day:14 month:01 pages:225-235 https://doi.org/10.1007/s00026-019-00478-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_267 AR 24 2020 2 14 01 225-235 |
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10.1007/s00026-019-00478-z doi (DE-627)OLC2061537642 (DE-He213)s00026-019-00478-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Lund, Ben verfasserin aut A Refined Energy Bound for Distinct Perpendicular Bisectors 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Nature Switzerland AG 2020 Abstract Let $${\mathcal {P}}$$ be a set of n points in the Euclidean plane. We prove that, for any $$\varepsilon > 0$$, either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors determined by pairs of points in $${\mathcal {P}}$$ is $$\Omega (n^{52/35 - \varepsilon })$$, where the constant implied by the $$\Omega $$ notation depends on $$\varepsilon $$. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors is $$\Omega (n^2)$$. The proof relies bounding the size of a carefully selected subset of the quadruples $$(a,b,c,d) \in {\mathcal {P}}^4$$ such that the perpendicular bisector of a and b is the same as the perpendicular bisector of c and d. Incidences Perpendicular bisectors Distinct distances Energy bound Enthalten in Annals of combinatorics Springer International Publishing, 1997 24(2020), 2 vom: 14. Jan., Seite 225-235 (DE-627)234146176 (DE-600)1394631-6 (DE-576)094078408 0218-0006 nnns volume:24 year:2020 number:2 day:14 month:01 pages:225-235 https://doi.org/10.1007/s00026-019-00478-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_267 AR 24 2020 2 14 01 225-235 |
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10.1007/s00026-019-00478-z doi (DE-627)OLC2061537642 (DE-He213)s00026-019-00478-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Lund, Ben verfasserin aut A Refined Energy Bound for Distinct Perpendicular Bisectors 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Nature Switzerland AG 2020 Abstract Let $${\mathcal {P}}$$ be a set of n points in the Euclidean plane. We prove that, for any $$\varepsilon > 0$$, either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors determined by pairs of points in $${\mathcal {P}}$$ is $$\Omega (n^{52/35 - \varepsilon })$$, where the constant implied by the $$\Omega $$ notation depends on $$\varepsilon $$. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors is $$\Omega (n^2)$$. The proof relies bounding the size of a carefully selected subset of the quadruples $$(a,b,c,d) \in {\mathcal {P}}^4$$ such that the perpendicular bisector of a and b is the same as the perpendicular bisector of c and d. Incidences Perpendicular bisectors Distinct distances Energy bound Enthalten in Annals of combinatorics Springer International Publishing, 1997 24(2020), 2 vom: 14. Jan., Seite 225-235 (DE-627)234146176 (DE-600)1394631-6 (DE-576)094078408 0218-0006 nnns volume:24 year:2020 number:2 day:14 month:01 pages:225-235 https://doi.org/10.1007/s00026-019-00478-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_267 AR 24 2020 2 14 01 225-235 |
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10.1007/s00026-019-00478-z doi (DE-627)OLC2061537642 (DE-He213)s00026-019-00478-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Lund, Ben verfasserin aut A Refined Energy Bound for Distinct Perpendicular Bisectors 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Nature Switzerland AG 2020 Abstract Let $${\mathcal {P}}$$ be a set of n points in the Euclidean plane. We prove that, for any $$\varepsilon > 0$$, either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors determined by pairs of points in $${\mathcal {P}}$$ is $$\Omega (n^{52/35 - \varepsilon })$$, where the constant implied by the $$\Omega $$ notation depends on $$\varepsilon $$. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors is $$\Omega (n^2)$$. The proof relies bounding the size of a carefully selected subset of the quadruples $$(a,b,c,d) \in {\mathcal {P}}^4$$ such that the perpendicular bisector of a and b is the same as the perpendicular bisector of c and d. Incidences Perpendicular bisectors Distinct distances Energy bound Enthalten in Annals of combinatorics Springer International Publishing, 1997 24(2020), 2 vom: 14. Jan., Seite 225-235 (DE-627)234146176 (DE-600)1394631-6 (DE-576)094078408 0218-0006 nnns volume:24 year:2020 number:2 day:14 month:01 pages:225-235 https://doi.org/10.1007/s00026-019-00478-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_267 AR 24 2020 2 14 01 225-235 |
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Abstract Let $${\mathcal {P}}$$ be a set of n points in the Euclidean plane. We prove that, for any $$\varepsilon > 0$$, either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors determined by pairs of points in $${\mathcal {P}}$$ is $$\Omega (n^{52/35 - \varepsilon })$$, where the constant implied by the $$\Omega $$ notation depends on $$\varepsilon $$. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors is $$\Omega (n^2)$$. The proof relies bounding the size of a carefully selected subset of the quadruples $$(a,b,c,d) \in {\mathcal {P}}^4$$ such that the perpendicular bisector of a and b is the same as the perpendicular bisector of c and d. © Springer Nature Switzerland AG 2020 |
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Abstract Let $${\mathcal {P}}$$ be a set of n points in the Euclidean plane. We prove that, for any $$\varepsilon > 0$$, either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors determined by pairs of points in $${\mathcal {P}}$$ is $$\Omega (n^{52/35 - \varepsilon })$$, where the constant implied by the $$\Omega $$ notation depends on $$\varepsilon $$. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors is $$\Omega (n^2)$$. The proof relies bounding the size of a carefully selected subset of the quadruples $$(a,b,c,d) \in {\mathcal {P}}^4$$ such that the perpendicular bisector of a and b is the same as the perpendicular bisector of c and d. © Springer Nature Switzerland AG 2020 |
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Abstract Let $${\mathcal {P}}$$ be a set of n points in the Euclidean plane. We prove that, for any $$\varepsilon > 0$$, either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors determined by pairs of points in $${\mathcal {P}}$$ is $$\Omega (n^{52/35 - \varepsilon })$$, where the constant implied by the $$\Omega $$ notation depends on $$\varepsilon $$. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors is $$\Omega (n^2)$$. The proof relies bounding the size of a carefully selected subset of the quadruples $$(a,b,c,d) \in {\mathcal {P}}^4$$ such that the perpendicular bisector of a and b is the same as the perpendicular bisector of c and d. © Springer Nature Switzerland AG 2020 |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">OLC2061537642</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230504152720.0</controlfield><controlfield tag="007">tu</controlfield><controlfield tag="008">200819s2020 xx ||||| 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s00026-019-00478-z</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)OLC2061537642</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-He213)s00026-019-00478-z-p</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="q">VZ</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">17,1</subfield><subfield code="2">ssgn</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Lund, Ben</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">A Refined Energy Bound for Distinct Perpendicular Bisectors</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2020</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">ohne Hilfsmittel zu benutzen</subfield><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Band</subfield><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© Springer Nature Switzerland AG 2020</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract Let $${\mathcal {P}}$$ be a set of n points in the Euclidean plane. We prove that, for any $$\varepsilon > 0$$, either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors determined by pairs of points in $${\mathcal {P}}$$ is $$\Omega (n^{52/35 - \varepsilon })$$, where the constant implied by the $$\Omega $$ notation depends on $$\varepsilon $$. This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains n/2 points of $${\mathcal {P}}$$, or the number of distinct perpendicular bisectors is $$\Omega (n^2)$$. The proof relies bounding the size of a carefully selected subset of the quadruples $$(a,b,c,d) \in {\mathcal {P}}^4$$ such that the perpendicular bisector of a and b is the same as the perpendicular bisector of c and d.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Incidences</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Perpendicular bisectors</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Distinct distances</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Energy bound</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Annals of combinatorics</subfield><subfield code="d">Springer International Publishing, 1997</subfield><subfield code="g">24(2020), 2 vom: 14. 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