A Generalization of the Convex Kakeya Problem

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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 I similar to(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.
Publisher
SPRINGER
Issue Date
2014-10
Language
English
Article Type
Article
Keywords

CURVES

Citation

ALGORITHMICA, v.70, no.2, pp.152 - 170

ISSN
0178-4617
DOI
10.1007/s00453-013-9831-y
URI
http://hdl.handle.net/10203/192354
Appears in Collection
CS-Journal Papers(저널논문)
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