TY - GEN

T1 - A generalization of the convex Kakeya problem

AU - Ahn, Heekap

AU - Bae, Sangwon

AU - Cheong, Otfried

AU - Gudmundsson, Joachim

AU - Tokuyama, Takeshi

AU - Vigneron, Antoine E.

N1 - KAUST Repository Item: Exported on 2020-10-01

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/575754

UR - http://link.springer.com/chapter/10.1007%2F978-3-642-29344-3_1

UR - http://www.scopus.com/inward/record.url?scp=84860821535&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-29344-3_1

DO - 10.1007/978-3-642-29344-3_1

M3 - Conference contribution

SN - 9783642293436

SP - 1

EP - 12

BT - LATIN 2012: Theoretical Informatics

PB - Springer Nature

ER -