A generalization of the convex Kakeya problem

Handle URI:
http://hdl.handle.net/10754/562979
Title:
A generalization of the convex Kakeya problem
Authors:
Ahn, Heekap; Bae, Sangwon; Cheong, Otfried; Gudmundsson, Joachim; Tokuyama, Takeshi; Vigneron, Antoine E. ( 0000-0003-3586-3431 )
Abstract:
Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal Θ(nlogn)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G. © 2013 Springer Science+Business Media New York.
KAUST Department:
Visual Computing Center (VCC); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Computer Science Program; Geometric Algorithms Group
Publisher:
Springer Science + Business Media
Journal:
Algorithmica
Issue Date:
19-Sep-2013
DOI:
10.1007/s00453-013-9831-y
ARXIV:
arXiv:1209.2171
Type:
Article
ISSN:
01784617
Sponsors:
H.-K.A. was supported by NRF grant 2011-0030044 (SRC-GAIA) funded by the government of Korea. J.G. is the recipient of an Australian Research Council Future Fellowship (project number FT100100755). S.W. Bae was supported by NRF grant (NRF-2013R1A1A1A05006927) funded by the government of Korea. O.C. was supported in part by NRF grant 2011-0030044 (SRC-GAIA), and in part by NRF grant 2011-0016434, both funded by the government of Korea.
Additional Links:
http://arxiv.org/abs/arXiv:1209.2171v1
Appears in Collections:
Articles; Computer Science Program; Visual Computing Center (VCC); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorAhn, Heekapen
dc.contributor.authorBae, Sangwonen
dc.contributor.authorCheong, Otfrieden
dc.contributor.authorGudmundsson, Joachimen
dc.contributor.authorTokuyama, Takeshien
dc.contributor.authorVigneron, Antoine E.en
dc.date.accessioned2015-08-03T11:17:53Zen
dc.date.available2015-08-03T11:17:53Zen
dc.date.issued2013-09-19en
dc.identifier.issn01784617en
dc.identifier.doi10.1007/s00453-013-9831-yen
dc.identifier.urihttp://hdl.handle.net/10754/562979en
dc.description.abstractGiven a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal Θ(nlogn)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G. © 2013 Springer Science+Business Media New York.en
dc.description.sponsorshipH.-K.A. was supported by NRF grant 2011-0030044 (SRC-GAIA) funded by the government of Korea. J.G. is the recipient of an Australian Research Council Future Fellowship (project number FT100100755). S.W. Bae was supported by NRF grant (NRF-2013R1A1A1A05006927) funded by the government of Korea. O.C. was supported in part by NRF grant 2011-0030044 (SRC-GAIA), and in part by NRF grant 2011-0016434, both funded by the government of Korea.en
dc.publisherSpringer Science + Business Mediaen
dc.relation.urlhttp://arxiv.org/abs/arXiv:1209.2171v1en
dc.subjectAlgorithmsen
dc.subjectComputational geometryen
dc.subjectDiscrete geometryen
dc.subjectKakeyaen
dc.titleA generalization of the convex Kakeya problemen
dc.typeArticleen
dc.contributor.departmentVisual Computing Center (VCC)en
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentComputer Science Programen
dc.contributor.departmentGeometric Algorithms Groupen
dc.identifier.journalAlgorithmicaen
dc.contributor.institutionPOSTECH, Pohang, South Koreaen
dc.contributor.institutionKyonggi University, Suwon, South Koreaen
dc.contributor.institutionKAIST, Daejeon, South Koreaen
dc.contributor.institutionUniversity of Sydney, NICTA, Sydney, Australiaen
dc.contributor.institutionTohoku University, Sendai, Japanen
dc.identifier.arxividarXiv:1209.2171en
kaust.authorVigneron, Antoine E.en
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