A generalization of the convex Kakeya problem

Hee Kap Ahn, Sang Won Bae, Otfried Cheong, Joachim Gudmundsson, Takeshi Tokuyama, Antoine Vigneron

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)152-170
Number of pages19
JournalAlgorithmica
Volume70
Issue number2
DOIs
Publication statusPublished - 2014 Oct

Keywords

  • Algorithms
  • Computational geometry
  • Discrete geometry
  • Kakeya

ASJC Scopus subject areas

  • Computer Science(all)
  • Computer Science Applications
  • Applied Mathematics

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