Abstract.
We consider the problem of bounding the complexity of the k th level in an arrangement of n curves or surfaces, a problem dual to, and an extension of, the well-known k-set problem. Among other results, we prove a new bound, O(nk 5/3 ) , on the complexity of the k th level in an arrangement of n planes in R 3 , or on the number of k -sets in a set of n points in three dimensions, and we show that the complexity of the k th level in an arrangement of n line segments in the plane is \(O(n\sqrt{k}\alpha(n/k))\) , and that the complexity of the k th level in an arrangement of n triangles in 3-space is O(n 2 k 5/6 α(n/k)) . <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p315.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader>
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Received February 7, 1997, and in revised form May 15, 1997, and August 30, 1997.
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Agarwal, P., Aronov, B., Chan, T. et al. On Levels in Arrangements of Lines, Segments, Planes, and Triangles% . Discrete Comput Geom 19, 315–331 (1998). https://doi.org/10.1007/PL00009348
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DOI: https://doi.org/10.1007/PL00009348