TY - GEN

T1 - A generalization of the convex Kakeya problem

AU - Ahn, Hee Kap

AU - Bae, Sang Won

AU - Cheong, Otfried

AU - Gudmundsson, Joachim

AU - Tokuyama, Takeshi

AU - Vigneron, Antoine

N1 - Funding Information:
H. A. was supported by NRF grant 2011-0030044 (SRC-GAIA) funded by the government of Korea. J.G. is the recipient of an Australian Research Council Future Fellowship (project number FT100100755). O.C. was supported in part by NRF grant 2011-0030044 (SRC-GAIA), and in part by NRF grant 2011-0016434, both funded by the government of Korea.

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.

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.

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

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

M3 - Conference contribution

AN - SCOPUS:84860821535

SN - 9783642293436

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 1

EP - 12

BT - LATIN 2012

T2 - 10th Latin American Symposiumon Theoretical Informatics, LATIN 2012

Y2 - 16 April 2012 through 20 April 2012

ER -