A Helly-type theorem for higher-dimensional transversals

Boris Aronov; Jacob E. Goodman; Richard Pollack

doi:10.1016/S0925-7721(01)00025-6

A Helly-type theorem for higher-dimensional transversals

Boris Aronov, Jacob E. Goodman, Richard Pollack

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

Abstract

We prove that a collection of compact convex sets of bounded diameters in K.d that is unbounded in k independent directions has a fc-flat transversal for k < d if and only if every d + 1 of the sets have a/:-transversal. This result generalizes a theorem of Hadwiger(-Danzer-Griinbaum-Klee) on line transversals for an unbounded family of compact convex sets. It is the first Helly-type theorem known for transversals of dimension between 1 and d -1.

Original language	English (US)
Pages (from-to)	177-183
Number of pages	7
Journal	Computational Geometry: Theory and Applications
Volume	21
Issue number	3
DOIs	https://doi.org/10.1016/S0925-7721(01)00025-6
State	Published - 2002

Keywords

-4-unbounded
Convex sets
Geometric transversal theory
Helly-type theorem
Jt-transversal

ASJC Scopus subject areas

Computer Science Applications
Geometry and Topology
Control and Optimization
Computational Theory and Mathematics
Computational Mathematics

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10.1016/S0925-7721(01)00025-6

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title = "A Helly-type theorem for higher-dimensional transversals",

abstract = "We prove that a collection of compact convex sets of bounded diameters in K.d that is unbounded in k independent directions has a fc-flat transversal for k < d if and only if every d + 1 of the sets have a/:-transversal. This result generalizes a theorem of Hadwiger(-Danzer-Griinbaum-Klee) on line transversals for an unbounded family of compact convex sets. It is the first Helly-type theorem known for transversals of dimension between 1 and d -1.",

keywords = "-4-unbounded, Convex sets, Geometric transversal theory, Helly-type theorem, Jt-transversal",

author = "Boris Aronov and Goodman, {Jacob E.} and Richard Pollack",

note = "Funding Information: *Corresponding author. E-mail addresses: [email protected] (B. Aronov), [email protected] (J.E. Goodman), [email protected] (R. Pollack). 1Supported in part by NSF Grant CCR-9972568. 2Supported in part by NSA Grants MDA904-98-I-0032 and MDA904-00-I-0013 and by PSC-CUNY Grants 61455-00-30 and 62495-00-31. 3Supported in part by NSF Grants CCR-9711240 and CCR-9732101 and NSA Grant MDA904-98-I-0505.",

year = "2002",

doi = "10.1016/S0925-7721(01)00025-6",

language = "English (US)",

volume = "21",

pages = "177--183",

journal = "Computational Geometry: Theory and Applications",

issn = "0925-7721",

publisher = "Elsevier",

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TY - JOUR

T1 - A Helly-type theorem for higher-dimensional transversals

AU - Aronov, Boris

AU - Goodman, Jacob E.

AU - Pollack, Richard

N1 - Funding Information: *Corresponding author. E-mail addresses: [email protected] (B. Aronov), [email protected] (J.E. Goodman), [email protected] (R. Pollack). 1Supported in part by NSF Grant CCR-9972568. 2Supported in part by NSA Grants MDA904-98-I-0032 and MDA904-00-I-0013 and by PSC-CUNY Grants 61455-00-30 and 62495-00-31. 3Supported in part by NSF Grants CCR-9711240 and CCR-9732101 and NSA Grant MDA904-98-I-0505.

PY - 2002

Y1 - 2002

N2 - We prove that a collection of compact convex sets of bounded diameters in K.d that is unbounded in k independent directions has a fc-flat transversal for k < d if and only if every d + 1 of the sets have a/:-transversal. This result generalizes a theorem of Hadwiger(-Danzer-Griinbaum-Klee) on line transversals for an unbounded family of compact convex sets. It is the first Helly-type theorem known for transversals of dimension between 1 and d -1.

AB - We prove that a collection of compact convex sets of bounded diameters in K.d that is unbounded in k independent directions has a fc-flat transversal for k < d if and only if every d + 1 of the sets have a/:-transversal. This result generalizes a theorem of Hadwiger(-Danzer-Griinbaum-Klee) on line transversals for an unbounded family of compact convex sets. It is the first Helly-type theorem known for transversals of dimension between 1 and d -1.

KW - -4-unbounded

KW - Convex sets

KW - Geometric transversal theory

KW - Helly-type theorem

KW - Jt-transversal

UR - http://www.scopus.com/inward/record.url?scp=33750727092&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750727092&partnerID=8YFLogxK

U2 - 10.1016/S0925-7721(01)00025-6

DO - 10.1016/S0925-7721(01)00025-6

M3 - Article

AN - SCOPUS:33750727092

SN - 0925-7721

VL - 21

SP - 177

EP - 183

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

IS - 3

ER -

A Helly-type theorem for higher-dimensional transversals

Abstract

Keywords

ASJC Scopus subject areas

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