A generalization of the convex Kakeya problem

Heekap Ahn, Sangwon Bae, Otfried Cheong, Joachim Gudmundsson, Takeshi Tokuyama, Antoine E. Vigneron

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

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.
Original languageEnglish (US)
Title of host publicationLATIN 2012: Theoretical Informatics
PublisherSpringer Nature
Pages1-12
Number of pages12
ISBN (Print)9783642293436
DOIs
StatePublished - 2012

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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