Tverberg theorems over discrete sets of points

J. A. De Loera; T. A. Hogan; F. Meunier; N. H. Mustafa

doi:10.1090/conm/764/15332

Tverberg theorems over discrete sets of points

J. A. De Loera
, T. A. Hogan
, F. Meunier
, N. H. Mustafa

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

This paper discusses Tverberg-type theorems with coordinate constraints (i.e., versions of these theorems where all points lie within a subset S ⊂ R^d and the intersection of convex hulls is required to have a non-empty intersection with S). We determine the m-Tverberg number, when m ≥ 3, of any discrete subset S of R² (a generalization of an unpublished result of J.-P. Doignon). We also present improvements on the upper bounds for the Tverberg numbers of Z³ and Z^j × R^k and an integer version of the well-known positive-fraction selection lemma of J. Pach.

Original language	English
Title of host publication	Polytopes and Discrete Geometry
Editors	Gabriel Cunningham, Mark Mixer, Egon Schulte
Publisher	American Mathematical Society
Pages	57-69
Number of pages	13
ISBN (Print)	9781470448974
DOIs	https://doi.org/10.1090/conm/764/15332
Publication status	Published - 1 Jan 2021
Event	Special Session on Polytopes and Discrete Geometry, 2018 - Boston, United States Duration: 21 Apr 2018 → 22 Apr 2018

Publication series

Name	Contemporary Mathematics
Volume	764
ISSN (Print)	0271-4132
ISSN (Electronic)	1098-3627

Conference

Conference	Special Session on Polytopes and Discrete Geometry, 2018
Country/Territory	United States
City	Boston
Period	21/04/18 → 22/04/18

Access to Document

10.1090/conm/764/15332

Cite this

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title = "Tverberg theorems over discrete sets of points",

abstract = "This paper discusses Tverberg-type theorems with coordinate constraints (i.e., versions of these theorems where all points lie within a subset S ⊂ Rd and the intersection of convex hulls is required to have a non-empty intersection with S). We determine the m-Tverberg number, when m ≥ 3, of any discrete subset S of R2 (a generalization of an unpublished result of J.-P. Doignon). We also present improvements on the upper bounds for the Tverberg numbers of Z3 and Zj × Rk and an integer version of the well-known positive-fraction selection lemma of J. Pach.",

author = "\{De Loera\}, \{J. A.\} and Hogan, \{T. A.\} and F. Meunier and Mustafa, \{N. H.\}",

note = "Publisher Copyright: {\textcopyright} 2021 American Mathematical Society.; Special Session on Polytopes and Discrete Geometry, 2018 ; Conference date: 21-04-2018 Through 22-04-2018",

year = "2021",

month = jan,

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doi = "10.1090/conm/764/15332",

language = "English",

isbn = "9781470448974",

series = "Contemporary Mathematics",

publisher = "American Mathematical Society",

pages = "57--69",

editor = "Gabriel Cunningham and Mark Mixer and Egon Schulte",

booktitle = "Polytopes and Discrete Geometry",

}

De Loera, JA, Hogan, TA, Meunier, F & Mustafa, NH 2021, Tverberg theorems over discrete sets of points. in G Cunningham, M Mixer & E Schulte (eds), Polytopes and Discrete Geometry. Contemporary Mathematics, vol. 764, American Mathematical Society, pp. 57-69, Special Session on Polytopes and Discrete Geometry, 2018, Boston, United States, 21/04/18. https://doi.org/10.1090/conm/764/15332

Tverberg theorems over discrete sets of points. / De Loera, J. A.; Hogan, T. A.; Meunier, F. et al.
Polytopes and Discrete Geometry. ed. / Gabriel Cunningham; Mark Mixer; Egon Schulte. American Mathematical Society, 2021. p. 57-69 (Contemporary Mathematics; Vol. 764).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Tverberg theorems over discrete sets of points

AU - De Loera, J. A.

AU - Hogan, T. A.

AU - Meunier, F.

AU - Mustafa, N. H.

PY - 2021/1/1

Y1 - 2021/1/1

N2 - This paper discusses Tverberg-type theorems with coordinate constraints (i.e., versions of these theorems where all points lie within a subset S ⊂ Rd and the intersection of convex hulls is required to have a non-empty intersection with S). We determine the m-Tverberg number, when m ≥ 3, of any discrete subset S of R2 (a generalization of an unpublished result of J.-P. Doignon). We also present improvements on the upper bounds for the Tverberg numbers of Z3 and Zj × Rk and an integer version of the well-known positive-fraction selection lemma of J. Pach.

AB - This paper discusses Tverberg-type theorems with coordinate constraints (i.e., versions of these theorems where all points lie within a subset S ⊂ Rd and the intersection of convex hulls is required to have a non-empty intersection with S). We determine the m-Tverberg number, when m ≥ 3, of any discrete subset S of R2 (a generalization of an unpublished result of J.-P. Doignon). We also present improvements on the upper bounds for the Tverberg numbers of Z3 and Zj × Rk and an integer version of the well-known positive-fraction selection lemma of J. Pach.

UR - https://www.scopus.com/pages/publications/85113677878

U2 - 10.1090/conm/764/15332

DO - 10.1090/conm/764/15332

M3 - Conference contribution

AN - SCOPUS:85113677878

SN - 9781470448974

T3 - Contemporary Mathematics

SP - 57

EP - 69

BT - Polytopes and Discrete Geometry

A2 - Cunningham, Gabriel

A2 - Mixer, Mark

A2 - Schulte, Egon

PB - American Mathematical Society

T2 - Special Session on Polytopes and Discrete Geometry, 2018

Y2 - 21 April 2018 through 22 April 2018

ER -

Tverberg theorems over discrete sets of points

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