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<head><title>proximum -- computes the proximum of the Polyhedron/Cone to a point in euclidian metric</title>
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<div><h1>proximum -- computes the proximum of the Polyhedron/Cone to a point in euclidian metric</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt> q = proximum(p,P) </tt><br/><tt>q = proximum(p,C)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>p</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, over <a href="../../Macaulay2Doc/html/___Z__Z.html" title="the class of all integers">ZZ</a> or <a href="../../Macaulay2Doc/html/___Q__Q.html" title="the class of all rational numbers">QQ</a> with only one column representing a point</span></li>
<li><span><tt>P</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span></span></li>
<li><span><tt>C</tt>, <span>a <a href="___Cone.html">convex rational cone</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>q</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, over <a href="../../Macaulay2Doc/html/___Q__Q.html" title="the class of all rational numbers">QQ</a> with only one column representing the closest point</span></li>
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<div class="single"><h2>Description</h2>
<div><p/>
For a point <tt>p</tt> and a Polyhedron <tt>P</tt> or a Cone <tt>C</tt>, <tt>proximum</tt>
computes the point in <tt>P</tt> or <tt>C</tt> with minimal euclidian distance to <tt>p</tt>.<table class="examples"><tr><td><pre>i1 : P = crossPolytope 3
o1 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 8
number of rays => 0
number of vertices => 6
o1 : Polyhedron</pre>
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<tr><td><pre>i2 : p = matrix {{1},{2},{3}}
o2 = | 1 |
| 2 |
| 3 |
3 1
o2 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i3 : q = proximum(p,P)
o3 = | 0 |
| 1 |
| 0 |
3 1
o3 : Matrix QQ <--- QQ</pre>
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<div class="waystouse"><h2>Ways to use <tt>proximum</tt> :</h2>
<ul><li>proximum(Matrix,Cone)</li>
<li>proximum(Matrix,Polyhedron)</li>
</ul>
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