A generalization of the convex kakeya problem

Authors:
Hee-Kap Ahn;Sang Won Bae;Otfried Cheong;Joachim Gudmundsson;Takeshi Tokuyama;Antoine Vigneron
Affiliations:
POSTECH, South Korea;Kyonggi University, South Korea;KAIST, South Korea;University of Sydney, Australia;Tohoku University, Japan;KAUST, Saudi Arabia
Venue:
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Year:
2012

Citing 3
Cited 0

Covering curves by translates of a convex set

American Mathematical Monthly
Geometric optimization and sums of algebraic functions

SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Aligning Two Convex Figures to Minimize Area or Perimeter

Algorithmica

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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 logn)-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.