# Random Approximation of Convex Bodies: Monotonicity of the Volumes of Random Tetrahedra

Ontology type: schema:ScholarlyArticle

### Article Info

DATE

2017-08-07

AUTHORS ABSTRACT

Choose uniform random points X1,⋯,Xn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1, \dots , X_n$$\end{document} in a given convex set and let conv[X1,⋯,Xn]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text { conv}}[X_1, \dots , X_n]$$\end{document} be their convex hull. It is shown that in dimension three the expected volume of this convex hull is in general not monotone with respect to set inclusion. This answers a question by Meckes in the negative. The given counterexample is formed by uniformly distributed points in the three-dimensional tetrahedron together with a small perturbation of it. As side result we obtain an explicit formula for all even moments of the volume of a random simplex which is the convex hull of three uniform random points in the tetrahedron and the center of one facet. More... »

PAGES

165-174

### References to SciGraph publications

• 2003-08. The Expected Volume of a Tetrahedron whose Vertices are Chosen at Random in the Interior of a Cube in MONATSHEFTE FÜR MATHEMATIK
• 2012-10-13. Random Polytopes in STOCHASTIC GEOMETRY, SPATIAL STATISTICS AND RANDOM FIELDS
• 2006-01-01. Disciplined Convex Programming in GLOBAL OPTIMIZATION
• 2008. Stochastic and Integral Geometry in NONE
• ### Journal

TITLE

Discrete & Computational Geometry

ISSUE

1

VOLUME

59

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00454-017-9914-7

DOI

http://dx.doi.org/10.1007/s00454-017-9914-7

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1091082566

Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service:

[
{
"@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json",
{
"id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01",
"inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/",
"name": "Mathematical Sciences",
"type": "DefinedTerm"
},
{
"id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0103",
"inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/",
"name": "Numerical and Computational Mathematics",
"type": "DefinedTerm"
}
],
"author": [
{
"affiliation": {
"alternateName": "Institut f\u00fcr Mathematik, Universit\u00e4t Osnabr\u00fcck, Albrechtstr. 28a, 49076, Osnabr\u00fcck, Germany",
"id": "http://www.grid.ac/institutes/grid.10854.38",
"name": [
"Institut f\u00fcr Mathematik, Universit\u00e4t Osnabr\u00fcck, Albrechtstr. 28a, 49076, Osnabr\u00fcck, Germany"
],
"type": "Organization"
},
"familyName": "Kunis",
"givenName": "Stefan",
"id": "sg:person.012526455753.27",
"sameAs": [
"https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012526455753.27"
],
"type": "Person"
},
{
"affiliation": {
"alternateName": "Fachbereich Mathematik, Universit\u00e4t Salzburg, Hellbrunnerstra\u00dfe 34, 5020, Salzburg, Austria",
"id": "http://www.grid.ac/institutes/grid.7039.d",
"name": [
"Fachbereich Mathematik, Universit\u00e4t Salzburg, Hellbrunnerstra\u00dfe 34, 5020, Salzburg, Austria"
],
"type": "Organization"
},
"familyName": "Reichenwallner",
"givenName": "Benjamin",
"id": "sg:person.011260700237.18",
"sameAs": [
"https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011260700237.18"
],
"type": "Person"
},
{
"affiliation": {
"alternateName": "Institut f\u00fcr Mathematik, Universit\u00e4t Osnabr\u00fcck, Albrechtstr. 28a, 49076, Osnabr\u00fcck, Germany",
"id": "http://www.grid.ac/institutes/grid.10854.38",
"name": [
"Institut f\u00fcr Mathematik, Universit\u00e4t Osnabr\u00fcck, Albrechtstr. 28a, 49076, Osnabr\u00fcck, Germany"
],
"type": "Organization"
},
"familyName": "Reitzner",
"givenName": "Matthias",
"id": "sg:person.010014726714.32",
"sameAs": [
"https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010014726714.32"
],
"type": "Person"
}
],
"citation": [
{
"id": "sg:pub.10.1007/978-3-642-33305-7_7",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1020747328",
"https://doi.org/10.1007/978-3-642-33305-7_7"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/s00605-002-0531-y",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1050777802",
"https://doi.org/10.1007/s00605-002-0531-y"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/978-3-540-78859-1",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1050491356",
"https://doi.org/10.1007/978-3-540-78859-1"
],
"type": "CreativeWork"
},
{
"id": "sg:pub.10.1007/0-387-30528-9_7",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1022540062",
"https://doi.org/10.1007/0-387-30528-9_7"
],
"type": "CreativeWork"
}
],
"datePublished": "2017-08-07",
"datePublishedReg": "2017-08-07",
"description": "Choose uniform random points X1,\u22ef,Xn\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X_1, \\dots , X_n$$\\end{document} in a given convex set and let conv[X1,\u22ef,Xn]\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text { conv}}[X_1, \\dots , X_n]$$\\end{document} be their convex hull. It is shown that in dimension three the expected volume of this convex hull is in general not monotone with respect to set inclusion. This answers a question by Meckes in the negative. The given counterexample is formed by uniformly distributed points in the three-dimensional tetrahedron together with a small perturbation of it. As side result we obtain an explicit formula for all even moments of the volume of a random simplex which is the convex hull of three uniform random points in the tetrahedron and the center of one facet.",
"genre": "article",
"id": "sg:pub.10.1007/s00454-017-9914-7",
"isAccessibleForFree": false,
"isPartOf": [
{
"id": "sg:journal.1043660",
"issn": [
"0179-5376",
"1432-0444"
],
"name": "Discrete & Computational Geometry",
"publisher": "Springer Nature",
"type": "Periodical"
},
{
"issueNumber": "1",
"type": "PublicationIssue"
},
{
"type": "PublicationVolume",
}
],
"keywords": [
"uniform random points",
"convex hull",
"random points",
"random approximation",
"convex sets",
"random simplex",
"dimension three",
"small perturbations",
"explicit formula",
"convex bodies",
"even moments",
"side result",
"hull",
"approximation",
"monotonicity",
"three-dimensional tetrahedra",
"point",
"counterexamples",
"tetrahedra",
"perturbations",
"Mecke",
"formula",
"moment",
"simplex",
"set",
"respect",
"three",
"results",
"volume",
"inclusion",
"questions",
"center",
"body",
"negatives",
"facets"
],
"name": "Random Approximation of Convex Bodies: Monotonicity of the Volumes of Random Tetrahedra",
"pagination": "165-174",
"productId": [
{
"name": "dimensions_id",
"type": "PropertyValue",
"value": [
"pub.1091082566"
]
},
{
"name": "doi",
"type": "PropertyValue",
"value": [
"10.1007/s00454-017-9914-7"
]
}
],
"sameAs": [
"https://doi.org/10.1007/s00454-017-9914-7",
"https://app.dimensions.ai/details/publication/pub.1091082566"
],
"sdDataset": "articles",
"sdDatePublished": "2022-12-01T06:35",
"sdPublisher": {
"name": "Springer Nature - SN SciGraph project",
"type": "Organization"
},
"sdSource": "s3://com-springernature-scigraph/baseset/20221201/entities/gbq_results/article/article_742.jsonl",
"type": "ScholarlyArticle",
"url": "https://doi.org/10.1007/s00454-017-9914-7"
}
]

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s00454-017-9914-7'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s00454-017-9914-7'

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00454-017-9914-7'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00454-017-9914-7'

This table displays all metadata directly associated to this object as RDF triples.

125 TRIPLES      21 PREDICATES      63 URIs      51 LITERALS      6 BLANK NODES

Subject Predicate Object
2 anzsrc-for:0103
3 schema:author N6e59efe6838d4e59bf7c4119a6c6619d
4 schema:citation sg:pub.10.1007/0-387-30528-9_7
5 sg:pub.10.1007/978-3-540-78859-1
6 sg:pub.10.1007/978-3-642-33305-7_7
7 sg:pub.10.1007/s00605-002-0531-y
8 schema:datePublished 2017-08-07
9 schema:datePublishedReg 2017-08-07
10 schema:description Choose uniform random points X1,⋯,Xn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1, \dots , X_n$$\end{document} in a given convex set and let conv[X1,⋯,Xn]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text { conv}}[X_1, \dots , X_n]$$\end{document} be their convex hull. It is shown that in dimension three the expected volume of this convex hull is in general not monotone with respect to set inclusion. This answers a question by Meckes in the negative. The given counterexample is formed by uniformly distributed points in the three-dimensional tetrahedron together with a small perturbation of it. As side result we obtain an explicit formula for all even moments of the volume of a random simplex which is the convex hull of three uniform random points in the tetrahedron and the center of one facet.
11 schema:genre article
12 schema:isAccessibleForFree false
13 schema:isPartOf N079afc82041b4f7997e0c6a0f6a3b49f
15 sg:journal.1043660
16 schema:keywords Mecke
17 approximation
18 body
19 center
20 convex bodies
21 convex hull
22 convex sets
23 counterexamples
24 dimension three
25 even moments
26 explicit formula
27 facets
28 formula
29 hull
30 inclusion
31 moment
32 monotonicity
33 negatives
34 perturbations
35 point
36 questions
37 random approximation
38 random points
39 random simplex
40 respect
41 results
42 set
43 side result
44 simplex
45 small perturbations
46 tetrahedra
47 three
48 three-dimensional tetrahedra
49 uniform random points
50 volume
51 schema:name Random Approximation of Convex Bodies: Monotonicity of the Volumes of Random Tetrahedra
52 schema:pagination 165-174
53 schema:productId N17ef5b712abd4674b9f8da8d8f856c44
54 N403f592944d347d389559a315f84c06d
55 schema:sameAs https://app.dimensions.ai/details/publication/pub.1091082566
56 https://doi.org/10.1007/s00454-017-9914-7
57 schema:sdDatePublished 2022-12-01T06:35
59 schema:sdPublisher Nef8b0453b59f40c6a812c0e6c91e9167
60 schema:url https://doi.org/10.1007/s00454-017-9914-7
62 sgo:sdDataset articles
63 rdf:type schema:ScholarlyArticle
64 N079afc82041b4f7997e0c6a0f6a3b49f schema:issueNumber 1
65 rdf:type schema:PublicationIssue
66 N17ef5b712abd4674b9f8da8d8f856c44 schema:name dimensions_id
67 schema:value pub.1091082566
68 rdf:type schema:PropertyValue
69 N403f592944d347d389559a315f84c06d schema:name doi
70 schema:value 10.1007/s00454-017-9914-7
71 rdf:type schema:PropertyValue
72 N6e59efe6838d4e59bf7c4119a6c6619d rdf:first sg:person.012526455753.27
73 rdf:rest Nc25994e95f0442c6a6c7d3b29deb9f24
75 rdf:type schema:PublicationVolume
76 Nc25994e95f0442c6a6c7d3b29deb9f24 rdf:first sg:person.011260700237.18
79 rdf:rest rdf:nil
80 Nef8b0453b59f40c6a812c0e6c91e9167 schema:name Springer Nature - SN SciGraph project
81 rdf:type schema:Organization
82 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
83 schema:name Mathematical Sciences
84 rdf:type schema:DefinedTerm
85 anzsrc-for:0103 schema:inDefinedTermSet anzsrc-for:
86 schema:name Numerical and Computational Mathematics
87 rdf:type schema:DefinedTerm
88 sg:journal.1043660 schema:issn 0179-5376
89 1432-0444
90 schema:name Discrete & Computational Geometry
91 schema:publisher Springer Nature
92 rdf:type schema:Periodical
93 sg:person.010014726714.32 schema:affiliation grid-institutes:grid.10854.38
94 schema:familyName Reitzner
95 schema:givenName Matthias
96 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010014726714.32
97 rdf:type schema:Person
98 sg:person.011260700237.18 schema:affiliation grid-institutes:grid.7039.d
99 schema:familyName Reichenwallner
100 schema:givenName Benjamin
101 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011260700237.18
102 rdf:type schema:Person
103 sg:person.012526455753.27 schema:affiliation grid-institutes:grid.10854.38
104 schema:familyName Kunis
105 schema:givenName Stefan
106 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012526455753.27
107 rdf:type schema:Person
108 sg:pub.10.1007/0-387-30528-9_7 schema:sameAs https://app.dimensions.ai/details/publication/pub.1022540062
109 https://doi.org/10.1007/0-387-30528-9_7
110 rdf:type schema:CreativeWork
111 sg:pub.10.1007/978-3-540-78859-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1050491356
112 https://doi.org/10.1007/978-3-540-78859-1
113 rdf:type schema:CreativeWork
114 sg:pub.10.1007/978-3-642-33305-7_7 schema:sameAs https://app.dimensions.ai/details/publication/pub.1020747328
115 https://doi.org/10.1007/978-3-642-33305-7_7
116 rdf:type schema:CreativeWork
117 sg:pub.10.1007/s00605-002-0531-y schema:sameAs https://app.dimensions.ai/details/publication/pub.1050777802
118 https://doi.org/10.1007/s00605-002-0531-y
119 rdf:type schema:CreativeWork
120 grid-institutes:grid.10854.38 schema:alternateName Institut für Mathematik, Universität Osnabrück, Albrechtstr. 28a, 49076, Osnabrück, Germany
121 schema:name Institut für Mathematik, Universität Osnabrück, Albrechtstr. 28a, 49076, Osnabrück, Germany
122 rdf:type schema:Organization
123 grid-institutes:grid.7039.d schema:alternateName Fachbereich Mathematik, Universität Salzburg, Hellbrunnerstraße 34, 5020, Salzburg, Austria
124 schema:name Fachbereich Mathematik, Universität Salzburg, Hellbrunnerstraße 34, 5020, Salzburg, Austria
125 rdf:type schema:Organization