On Weak ϵ -Nets and the Radon Number

Shay Moran; Amir Yehudayoff

doi:https://doi.org/10.1007/s00454-020-00222-y

On Weak ϵ -Nets and the Radon Number

Shay Moran, Amir Yehudayoff

Technion - Israel Institute of Technology

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

Abstract

We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the Euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly’s property and on metric properties of VC classes. The lower bound on the size of weak nets when the Radon number is large relies on the chromatic number of the Kneser graph. As an application, we prove an amplification result for weak ϵ-nets.

Original language	English
Pages (from-to)	1125-1140
Number of pages	16
Journal	Discrete and Computational Geometry
Volume	64
Issue number	4
DOIs	https://doi.org/10.1007/s00454-020-00222-y
State	Published - Dec 2020

Keywords

Abstract convexity
Epsilon-nets
Helly number
Radon number

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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https://doi.org/10.1007/s00454-020-00222-y

Cite this

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title = "On Weak ϵ -Nets and the Radon Number",

abstract = "We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the Euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly{\textquoteright}s property and on metric properties of VC classes. The lower bound on the size of weak nets when the Radon number is large relies on the chromatic number of the Kneser graph. As an application, we prove an amplification result for weak ϵ-nets.",

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AU - Yehudayoff, Amir

PY - 2020/12

Y1 - 2020/12

N2 - We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the Euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly’s property and on metric properties of VC classes. The lower bound on the size of weak nets when the Radon number is large relies on the chromatic number of the Kneser graph. As an application, we prove an amplification result for weak ϵ-nets.

AB - We show that the Radon number characterizes the existence of weak nets in separable convexity spaces (an abstraction of the Euclidean notion of convexity). The construction of weak nets when the Radon number is finite is based on Helly’s property and on metric properties of VC classes. The lower bound on the size of weak nets when the Radon number is large relies on the chromatic number of the Kneser graph. As an application, we prove an amplification result for weak ϵ-nets.

KW - Abstract convexity

KW - Epsilon-nets

KW - Helly number

KW - Radon number

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DO - https://doi.org/10.1007/s00454-020-00222-y

M3 - مقالة

SN - 0179-5376

VL - 64

SP - 1125

EP - 1140

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 4

ER -

On Weak ϵ -Nets and the Radon Number

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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