Theorems of Carathéodory, Helly, and Tverberg Without Dimension

Karim Adiprasito; Imre Bárány; Nabil H. Mustafa; Tamás Terpai

doi:10.1007/s00454-020-00172-5

Theorems of Carathéodory, Helly, and Tverberg Without Dimension

Karim Adiprasito^*, Imre Bárány, Nabil H. Mustafa, Tamás Terpai

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We initiate the study of no-dimensional versions of classical theorems in convexity. One example is Carathéodory’s theorem without dimension: given an n-element set P in a Euclidean space, a point a∈convP, and an integer r≤ n, there is a subset Q⊂ P of r elements such that the distance between a and convQ is less than diamP/2r. In an analoguos no-dimension Helly theorem a finite family F of convex bodies is given, all of them are contained in the Euclidean unit ball of R^d. If k≤ d, | F| ≥ k, and every k-element subfamily of F is intersecting, then there is a point q∈ R^d which is closer than 1/k to every set in F. This result has several colourful and fractional consequences. Similar versions of Tverberg’s theorem and some of their extensions are also established.

Original language	American English
Pages (from-to)	233-258
Number of pages	26
Journal	Discrete and Computational Geometry
Volume	64
Issue number	2
DOIs	https://doi.org/10.1007/s00454-020-00172-5
State	Published - 1 Sep 2020

Bibliographical note

Funding Information:
K.A. was supported by ERC StG 716424-CASe and ISF Grant 1050/16. I.B. was supported by the Hungarian National Research, Development and Innovation Office NKFIH Grants K 111827 and K 116769, and by ERC-AdG 321104. N.M. was supported by the grant ANR SAGA (JCJC-14-CE25-0016-01). T.T. was supported by the Hungarian National Research, Development and Innovation Office NKFIH Grants NK 112735 and K 120697.

Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

Carathéodory theorem
Convex approximation
Tverberg theorem

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10.1007/s00454-020-00172-5

Cite this

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title = "Theorems of Carath{\'e}odory, Helly, and Tverberg Without Dimension",

abstract = "We initiate the study of no-dimensional versions of classical theorems in convexity. One example is Carath{\'e}odory{\textquoteright}s theorem without dimension: given an n-element set P in a Euclidean space, a point a∈convP, and an integer r≤ n, there is a subset Q⊂ P of r elements such that the distance between a and convQ is less than diamP/2r. In an analoguos no-dimension Helly theorem a finite family F of convex bodies is given, all of them are contained in the Euclidean unit ball of Rd. If k≤ d, | F| ≥ k, and every k-element subfamily of F is intersecting, then there is a point q∈ Rd which is closer than 1/k to every set in F. This result has several colourful and fractional consequences. Similar versions of Tverberg{\textquoteright}s theorem and some of their extensions are also established.",

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author = "Karim Adiprasito and Imre B{\'a}r{\'a}ny and Mustafa, {Nabil H.} and Tam{\'a}s Terpai",

note = "Funding Information: K.A. was supported by ERC StG 716424-CASe and ISF Grant 1050/16. I.B. was supported by the Hungarian National Research, Development and Innovation Office NKFIH Grants K 111827 and K 116769, and by ERC-AdG 321104. N.M. was supported by the grant ANR SAGA (JCJC-14-CE25-0016-01). T.T. was supported by the Hungarian National Research, Development and Innovation Office NKFIH Grants NK 112735 and K 120697. Publisher Copyright: {\textcopyright} 2020, Springer Science+Business Media, LLC, part of Springer Nature.",

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AU - Terpai, Tamás

N1 - Funding Information: K.A. was supported by ERC StG 716424-CASe and ISF Grant 1050/16. I.B. was supported by the Hungarian National Research, Development and Innovation Office NKFIH Grants K 111827 and K 116769, and by ERC-AdG 321104. N.M. was supported by the grant ANR SAGA (JCJC-14-CE25-0016-01). T.T. was supported by the Hungarian National Research, Development and Innovation Office NKFIH Grants NK 112735 and K 120697. Publisher Copyright: © 2020, Springer Science+Business Media, LLC, part of Springer Nature.

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N2 - We initiate the study of no-dimensional versions of classical theorems in convexity. One example is Carathéodory’s theorem without dimension: given an n-element set P in a Euclidean space, a point a∈convP, and an integer r≤ n, there is a subset Q⊂ P of r elements such that the distance between a and convQ is less than diamP/2r. In an analoguos no-dimension Helly theorem a finite family F of convex bodies is given, all of them are contained in the Euclidean unit ball of Rd. If k≤ d, | F| ≥ k, and every k-element subfamily of F is intersecting, then there is a point q∈ Rd which is closer than 1/k to every set in F. This result has several colourful and fractional consequences. Similar versions of Tverberg’s theorem and some of their extensions are also established.

AB - We initiate the study of no-dimensional versions of classical theorems in convexity. One example is Carathéodory’s theorem without dimension: given an n-element set P in a Euclidean space, a point a∈convP, and an integer r≤ n, there is a subset Q⊂ P of r elements such that the distance between a and convQ is less than diamP/2r. In an analoguos no-dimension Helly theorem a finite family F of convex bodies is given, all of them are contained in the Euclidean unit ball of Rd. If k≤ d, | F| ≥ k, and every k-element subfamily of F is intersecting, then there is a point q∈ Rd which is closer than 1/k to every set in F. This result has several colourful and fractional consequences. Similar versions of Tverberg’s theorem and some of their extensions are also established.

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Theorems of Carathéodory, Helly, and Tverberg Without Dimension

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Keywords

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