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

Mathematics

Technion - Israel Institute of Technology

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

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	American 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

Keywords

Cubic curves
Elliptic curves
Line blocker
Lines
Points

All Science Journal Classification (ASJC) codes

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

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10.1007/s00454-020-00201-3

Cite this

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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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JF - Discrete and Computational Geometry

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ER -

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

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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