Spherical convex hull of random points on a wedge
Consider two half-spaces H 1 + and H 2 + in R d + 1 whose bounding hyperplanes H 1 and H 2 are orthogonal and pass through the origin. The intersection S 2 , + d : = S d ∩ H 1 + ∩ H 2 + is a spherical convex subset of the d -dimensional unit sphere S d , which contains a great subsphere of dimension...
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| Veröffentlicht in: | Mathematische annalen 2024-07, Vol.389 (3), p.2289-2316 |
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| container_title | Mathematische annalen |
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| creator | Besau, Florian Gusakova, Anna Reitzner, Matthias Schütt, Carsten Thäle, Christoph Werner, Elisabeth M. |
| description | Consider two half-spaces
H
1
+
and
H
2
+
in
R
d
+
1
whose bounding hyperplanes
H
1
and
H
2
are orthogonal and pass through the origin. The intersection
S
2
,
+
d
:
=
S
d
∩
H
1
+
∩
H
2
+
is a spherical convex subset of the
d
-dimensional unit sphere
S
d
, which contains a great subsphere of dimension
d
-
2
and is called a spherical wedge. Choose
n
independent random points uniformly at random on
S
2
,
+
d
and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of
log
n
. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on
S
2
,
+
d
. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere. |
| doi_str_mv | 10.1007/s00208-023-02704-9 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_3062683669</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>3062683669</sourcerecordid><originalsourceid>FETCH-LOGICAL-c314t-de27ac7c527041efc0fb1e948ea6faa9e21c8098178760eb068bcba61ee5f3d63</originalsourceid><addsrcrecordid>eNp9kEtPwzAQhC0EEqXwBzhZ4mxY24ntHFEFFKkSB-BsOc66D6VxsFse_56UIHHjsNrDzsyOPkIuOVxzAH2TAQQYBkIOo6Fg1RGZ8EIKxg3oYzIZ7iUrjeSn5CznDQBIgHJC-HO_wrT2rqU-du_4SVf7tqUx0OS6Jm5pH9fdLtPYUUc_sFniOTkJrs148bun5PX-7mU2Z4unh8fZ7YJ5yYsda1Bo57UvD3U4Bg-h5lgVBp0KzlUouDdQGa6NVoA1KFP72imOWAbZKDklV2Nun-LbHvPObuI-dcNLK0EJZaRS1aASo8qnmHPCYPu03rr0ZTnYAxo7orEDGvuDxh5McjTlQdwtMf1F_-P6BlNOZYU</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>3062683669</pqid></control><display><type>article</type><title>Spherical convex hull of random points on a wedge</title><source>Springer Nature - Complete Springer Journals</source><creator>Besau, Florian ; Gusakova, Anna ; Reitzner, Matthias ; Schütt, Carsten ; Thäle, Christoph ; Werner, Elisabeth M.</creator><creatorcontrib>Besau, Florian ; Gusakova, Anna ; Reitzner, Matthias ; Schütt, Carsten ; Thäle, Christoph ; Werner, Elisabeth M.</creatorcontrib><description>Consider two half-spaces
H
1
+
and
H
2
+
in
R
d
+
1
whose bounding hyperplanes
H
1
and
H
2
are orthogonal and pass through the origin. The intersection
S
2
,
+
d
:
=
S
d
∩
H
1
+
∩
H
2
+
is a spherical convex subset of the
d
-dimensional unit sphere
S
d
, which contains a great subsphere of dimension
d
-
2
and is called a spherical wedge. Choose
n
independent random points uniformly at random on
S
2
,
+
d
and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of
log
n
. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on
S
2
,
+
d
. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.</description><identifier>ISSN: 0025-5831</identifier><identifier>EISSN: 1432-1807</identifier><identifier>DOI: 10.1007/s00208-023-02704-9</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Convexity ; Half spaces ; Hyperplanes ; Mathematics ; Mathematics and Statistics ; Polytopes</subject><ispartof>Mathematische annalen, 2024-07, Vol.389 (3), p.2289-2316</ispartof><rights>The Author(s) 2023</rights><rights>The Author(s) 2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c314t-de27ac7c527041efc0fb1e948ea6faa9e21c8098178760eb068bcba61ee5f3d63</cites><orcidid>0000-0002-7922-6429</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00208-023-02704-9$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00208-023-02704-9$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,27903,27904,41467,42536,51298</link.rule.ids></links><search><creatorcontrib>Besau, Florian</creatorcontrib><creatorcontrib>Gusakova, Anna</creatorcontrib><creatorcontrib>Reitzner, Matthias</creatorcontrib><creatorcontrib>Schütt, Carsten</creatorcontrib><creatorcontrib>Thäle, Christoph</creatorcontrib><creatorcontrib>Werner, Elisabeth M.</creatorcontrib><title>Spherical convex hull of random points on a wedge</title><title>Mathematische annalen</title><addtitle>Math. Ann</addtitle><description>Consider two half-spaces
H
1
+
and
H
2
+
in
R
d
+
1
whose bounding hyperplanes
H
1
and
H
2
are orthogonal and pass through the origin. The intersection
S
2
,
+
d
:
=
S
d
∩
H
1
+
∩
H
2
+
is a spherical convex subset of the
d
-dimensional unit sphere
S
d
, which contains a great subsphere of dimension
d
-
2
and is called a spherical wedge. Choose
n
independent random points uniformly at random on
S
2
,
+
d
and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of
log
n
. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on
S
2
,
+
d
. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.</description><subject>Convexity</subject><subject>Half spaces</subject><subject>Hyperplanes</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Polytopes</subject><issn>0025-5831</issn><issn>1432-1807</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp9kEtPwzAQhC0EEqXwBzhZ4mxY24ntHFEFFKkSB-BsOc66D6VxsFse_56UIHHjsNrDzsyOPkIuOVxzAH2TAQQYBkIOo6Fg1RGZ8EIKxg3oYzIZ7iUrjeSn5CznDQBIgHJC-HO_wrT2rqU-du_4SVf7tqUx0OS6Jm5pH9fdLtPYUUc_sFniOTkJrs148bun5PX-7mU2Z4unh8fZ7YJ5yYsda1Bo57UvD3U4Bg-h5lgVBp0KzlUouDdQGa6NVoA1KFP72imOWAbZKDklV2Nun-LbHvPObuI-dcNLK0EJZaRS1aASo8qnmHPCYPu03rr0ZTnYAxo7orEDGvuDxh5McjTlQdwtMf1F_-P6BlNOZYU</recordid><startdate>20240701</startdate><enddate>20240701</enddate><creator>Besau, Florian</creator><creator>Gusakova, Anna</creator><creator>Reitzner, Matthias</creator><creator>Schütt, Carsten</creator><creator>Thäle, Christoph</creator><creator>Werner, Elisabeth M.</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-7922-6429</orcidid></search><sort><creationdate>20240701</creationdate><title>Spherical convex hull of random points on a wedge</title><author>Besau, Florian ; Gusakova, Anna ; Reitzner, Matthias ; Schütt, Carsten ; Thäle, Christoph ; Werner, Elisabeth M.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c314t-de27ac7c527041efc0fb1e948ea6faa9e21c8098178760eb068bcba61ee5f3d63</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Convexity</topic><topic>Half spaces</topic><topic>Hyperplanes</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Polytopes</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Besau, Florian</creatorcontrib><creatorcontrib>Gusakova, Anna</creatorcontrib><creatorcontrib>Reitzner, Matthias</creatorcontrib><creatorcontrib>Schütt, Carsten</creatorcontrib><creatorcontrib>Thäle, Christoph</creatorcontrib><creatorcontrib>Werner, Elisabeth M.</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><jtitle>Mathematische annalen</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Besau, Florian</au><au>Gusakova, Anna</au><au>Reitzner, Matthias</au><au>Schütt, Carsten</au><au>Thäle, Christoph</au><au>Werner, Elisabeth M.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Spherical convex hull of random points on a wedge</atitle><jtitle>Mathematische annalen</jtitle><stitle>Math. Ann</stitle><date>2024-07-01</date><risdate>2024</risdate><volume>389</volume><issue>3</issue><spage>2289</spage><epage>2316</epage><pages>2289-2316</pages><issn>0025-5831</issn><eissn>1432-1807</eissn><abstract>Consider two half-spaces
H
1
+
and
H
2
+
in
R
d
+
1
whose bounding hyperplanes
H
1
and
H
2
are orthogonal and pass through the origin. The intersection
S
2
,
+
d
:
=
S
d
∩
H
1
+
∩
H
2
+
is a spherical convex subset of the
d
-dimensional unit sphere
S
d
, which contains a great subsphere of dimension
d
-
2
and is called a spherical wedge. Choose
n
independent random points uniformly at random on
S
2
,
+
d
and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of
log
n
. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on
S
2
,
+
d
. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00208-023-02704-9</doi><tpages>28</tpages><orcidid>https://orcid.org/0000-0002-7922-6429</orcidid><oa>free_for_read</oa></addata></record> |
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| ispartof | Mathematische annalen, 2024-07, Vol.389 (3), p.2289-2316 |
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| language | eng |
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| source | Springer Nature - Complete Springer Journals |
| subjects | Convexity Half spaces Hyperplanes Mathematics Mathematics and Statistics Polytopes |
| title | Spherical convex hull of random points on a wedge |
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