Discrete and lexicographic Helly-type theorems

Nir Halman

doi:10.1007/s00454-007-9028-8

Discrete and lexicographic Helly-type theorems

Nir Halman

Massachusetts Institute of Technology

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

5 Scopus citations

Abstract

Helly's theorem says that if every d+1 elements of a given finite set of convex objects in ℝ^d have a common point, then there is a point common to all of the objects in the set. We define three new types of Helly theorems: discrete Helly theorems-where the common point should belong to an a-priori given set, lexicographic Helly theorems-where the common point should not be lexicographically greater than a given point, and lexicographic-discrete Helly theorems. We study the relations between the different types of the Helly theorems. We obtain several new discrete and lexicographic Helly numbers.

Original language	English
Pages (from-to)	690-719
Number of pages	30
Journal	Discrete and Computational Geometry
Volume	39
Issue number	4
DOIs	https://doi.org/10.1007/s00454-007-9028-8
State	Published - Jun 2008
Externally published	Yes

Keywords

Discrete geometry
Helly theorems

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10.1007/s00454-007-9028-8

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AB - Helly's theorem says that if every d+1 elements of a given finite set of convex objects in ℝd have a common point, then there is a point common to all of the objects in the set. We define three new types of Helly theorems: discrete Helly theorems-where the common point should belong to an a-priori given set, lexicographic Helly theorems-where the common point should not be lexicographically greater than a given point, and lexicographic-discrete Helly theorems. We study the relations between the different types of the Helly theorems. We obtain several new discrete and lexicographic Helly numbers.

KW - Discrete geometry

KW - Helly theorems

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JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 4

ER -

Discrete and lexicographic Helly-type theorems

Abstract

Keywords

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