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<H1>cutting theorem</H1>
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(definition)
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<strong>Definition:</strong>
For any set <strong><em>H</em></strong> of n hyperplanes in <em>R</em><sup>k</sup>, and any parameter r, 1 <img src="leq.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/leq.gif" border=0 height=7 width=14 alt="less than or equal to"> r <img src="leq.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/leq.gif" border=0 height=7 width=14 alt="less than or equal to"> n, there always exists a (1/r)-cutting of size O(r<sup>k</sup>). In two dimensions, a (1/r)-cutting of size s is a partition of the plane into s disjoint triangles, some of which are unbounded, such that no triangle in the partition intersects more than n/r lines in <strong><em>H</em></strong>. In <em>R</em><sup>k</sup>, triangles are replaced by <a href="#simplices"><em>simplices</em></a>. Such a cutting can be computed in O(nr<sup>k-1</sup>) time.
<P><em>Note:
From Algorithms and Theory of Computation Handbook, page 20-25, Copyright © 1999 by CRC Press LLC.</em>
<P>Author: <a href="terms.html#authorCRC-A" tppabs="http://hissa.nist.gov/dads/terms.html#authorCRC-A">CRC-A</a>
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Entry modified Thu Jun 17 09:56:40 1999.<BR>
HTML page formatted Wed Dec 22 09:35:04 1999.
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