Abstract.
We prove an O(n(k+1) 1/3 ) upper bound for planar k -sets. This is the first considerable improvement on this bound after its early solution approximately 27 years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of k -levels in the arrangement of line segments, k convex polygons in the union of n lines, parametric minimum spanning trees, and parametric matroids in general. <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>19n3p373.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader>
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Received May 31, 1997, and in revised form June 30, 1997.
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Dey, T. Improved Bounds for Planar k -Sets and Related Problems . Discrete Comput Geom 19, 373–382 (1998). https://doi.org/10.1007/PL00009354
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DOI: https://doi.org/10.1007/PL00009354