A generalization of the convex Kakeya problem

Handle URI:
http://hdl.handle.net/10754/575754
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:
We consider the following geometric alignment problem: Given a set of line segments in the plane, find a convex region of smallest area that contains a translate of each input segment. This can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be turned through 360 degrees within this region. Our main result is an optimal Θ(n log n)-time algorithm for our geometric alignment problem, when the input is a set of n line segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then the optimum placement is when the midpoints of the segments coincide. 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 any rotated copy of G. © 2012 Springer-Verlag Berlin Heidelberg.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Computer Science Program; Visual Computing Center (VCC); Geometric Algorithms Group
Citation:
Ahn HK., Bae S.W., Cheong O., Gudmundsson J., Tokuyama T., Vigneron A. (2012) A Generalization of the Convex Kakeya Problem. In: Fernández-Baca D. (eds) LATIN 2012: Theoretical Informatics. LATIN 2012. Lecture Notes in Computer Science, vol 7256. Springer, Berlin, Heidelberg
Publisher:
Springer Science + Business Media
Journal:
LATIN 2012: Theoretical Informatics
Conference/Event name:
10th Latin American Symposiumon Theoretical Informatics, LATIN 2012
Issue Date:
2012
DOI:
10.1007/978-3-642-29344-3_1
ARXIV:
arXiv:1209.2171v1
Type:
Conference Paper
ISSN:
03029743
ISBN:
9783642293436
Additional Links:
http://link.springer.com/chapter/10.1007%2F978-3-642-29344-3_1
Appears in Collections:
Conference Papers; Computer Science Program; Computer Science Program; Visual Computing Center (VCC); Visual Computing Center (VCC); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; 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.accessioned2015-08-24T09:25:19Zen
dc.date.available2015-08-24T09:25:19Zen
dc.date.issued2012en
dc.identifier.citationAhn HK., Bae S.W., Cheong O., Gudmundsson J., Tokuyama T., Vigneron A. (2012) A Generalization of the Convex Kakeya Problem. In: Fernández-Baca D. (eds) LATIN 2012: Theoretical Informatics. LATIN 2012. Lecture Notes in Computer Science, vol 7256. Springer, Berlin, Heidelbergen
dc.identifier.isbn9783642293436en
dc.identifier.issn03029743en
dc.identifier.doi10.1007/978-3-642-29344-3_1en
dc.identifier.urihttp://hdl.handle.net/10754/575754en
dc.description.abstractWe consider the following geometric alignment problem: Given a set of line segments in the plane, find a convex region of smallest area that contains a translate of each input segment. This can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be turned through 360 degrees within this region. Our main result is an optimal Θ(n log n)-time algorithm for our geometric alignment problem, when the input is a set of n line segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then the optimum placement is when the midpoints of the segments coincide. 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 any rotated copy of G. © 2012 Springer-Verlag Berlin Heidelberg.en
dc.publisherSpringer Science + Business Mediaen
dc.relation.urlhttp://link.springer.com/chapter/10.1007%2F978-3-642-29344-3_1en
dc.titleA generalization of the convex Kakeya problemen
dc.typeConference Paperen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentComputer Science Programen
dc.contributor.departmentVisual Computing Center (VCC)en
dc.contributor.departmentGeometric Algorithms Groupen
dc.identifier.journalLATIN 2012: Theoretical Informaticsen
dc.conference.date16 April 2012 through 20 April 2012en
dc.conference.name10th Latin American Symposiumon Theoretical Informatics, LATIN 2012en
dc.conference.locationArequipaen
dc.contributor.institutionPOSTECH, South Koreaen
dc.contributor.institutionKyonggi University, South Koreaen
dc.contributor.institutionKAIST, South Koreaen
dc.contributor.institutionUniversity of Sydney, Australiaen
dc.contributor.institutionTohoku University, Japanen
dc.identifier.arxividarXiv:1209.2171v1en
kaust.authorVigneron, Antoine E.en
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