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<head><title>smallestFace -- determines the smallest face of the Cone/Polyhedron containing a point</title>
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<div><h1>smallestFace -- determines the smallest face of the Cone/Polyhedron containing a point</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> C1 = smallestFace(p,C) </tt><br/><tt>P1 = smallestFace(p,P)</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>C</tt>, <span>a <a href="___Cone.html">convex rational cone</a></span></span></li>
<li><span><tt>P</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>C1</tt>, <span>a <a href="___Cone.html">convex rational cone</a></span>, or</span></li>
<li><span><tt>P1</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span></span></li>
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<div class="single"><h2>Description</h2>
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<tt>p</tt> is considered to be a point in the ambient space of the second argument, so
the number of rows of <tt>p</tt> must equal the dimension of the ambient space of the
second argument. The function computes the smallest face of the second argument that
contains <tt>p</tt>. If the second argument is a <a href="___Polyhedron.html" title="the class of all convex polyhedra">Polyhedron</a> the output is a
<a href="___Polyhedron.html" title="the class of all convex polyhedra">Polyhedron</a> and if it is a <a href="___Cone.html" title="the class of all rational convex polyhedral cones">Cone</a> the output is a <a href="___Cone.html" title="the class of all rational convex polyhedral cones">Cone</a>. In both cases,
if the point is not contained in the second argument then the output is the empty
polyhedron.<table class="examples"><tr><td><pre>i1 : P = hypercube 3
o1 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 6
number of rays => 0
number of vertices => 8
o1 : Polyhedron</pre>
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<tr><td><pre>i2 : p = matrix {{1},{0},{0}}
o2 = | 1 |
| 0 |
| 0 |
3 1
o2 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i3 : smallestFace(p,P)
o3 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 4
number of rays => 0
number of vertices => 4
o3 : Polyhedron</pre>
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<div class="waystouse"><h2>Ways to use <tt>smallestFace</tt> :</h2>
<ul><li>smallestFace(Matrix,Cone)</li>
<li>smallestFace(Matrix,Polyhedron)</li>
</ul>
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