On Sets of n Points in General Position That Determine Lines That Can Be Pierced by n Points

Chaya Keller; Rom Pinchasi

doi:10.1007/s00454-020-00201-3

On Sets of n Points in General Position That Determine Lines That Can Be Pierced by n Points

Chaya Keller, Rom Pinchasi

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

2 Scopus citations

Abstract

Let P be a set of n points in general position in the plane. Let R be a set of n points disjoint from P such that for every x, y∈ P the line through x and y contains a point in R outside of the segment delimited by x and y. We show that P∪ R must be contained in cubic curve. This resolves a special case of a conjecture of Milićević. We use the same approach to solve a special case of a problem of Karasev related to a bipartite version of the above problem.

Original language	English
Pages (from-to)	905-915
Number of pages	11
Journal	Discrete and Computational Geometry
Volume	64
Issue number	3
DOIs	https://doi.org/10.1007/s00454-020-00201-3
State	Published - 1 Oct 2020
Externally published	Yes

Keywords

Cubic curves
Elliptic curves
Line blocker
Lines
Points

ASJC Scopus subject areas

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

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10.1007/s00454-020-00201-3

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abstract = "Let P be a set of n points in general position in the plane. Let R be a set of n points disjoint from P such that for every x, y∈ P the line through x and y contains a point in R outside of the segment delimited by x and y. We show that P∪ R must be contained in cubic curve. This resolves a special case of a conjecture of Mili{\'c}evi{\'c}. We use the same approach to solve a special case of a problem of Karasev related to a bipartite version of the above problem.",

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N2 - Let P be a set of n points in general position in the plane. Let R be a set of n points disjoint from P such that for every x, y∈ P the line through x and y contains a point in R outside of the segment delimited by x and y. We show that P∪ R must be contained in cubic curve. This resolves a special case of a conjecture of Milićević. We use the same approach to solve a special case of a problem of Karasev related to a bipartite version of the above problem.

AB - Let P be a set of n points in general position in the plane. Let R be a set of n points disjoint from P such that for every x, y∈ P the line through x and y contains a point in R outside of the segment delimited by x and y. We show that P∪ R must be contained in cubic curve. This resolves a special case of a conjecture of Milićević. We use the same approach to solve a special case of a problem of Karasev related to a bipartite version of the above problem.

KW - Cubic curves

KW - Elliptic curves

KW - Line blocker

KW - Lines

KW - Points

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IS - 3

ER -

On Sets of n Points in General Position That Determine Lines That Can Be Pierced by n Points

Abstract

Keywords

ASJC Scopus subject areas

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