TY - JOUR

T1 - Discrete and lexicographic Helly-type theorems

AU - Halman, Nir

PY - 2008/6

Y1 - 2008/6

N2 - 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.

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

UR - http://www.scopus.com/inward/record.url?scp=45749153868&partnerID=8YFLogxK

U2 - 10.1007/s00454-007-9028-8

DO - 10.1007/s00454-007-9028-8

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AN - SCOPUS:45749153868

SN - 0179-5376

VL - 39

SP - 690

EP - 719

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 4

ER -