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<p><a id="X8524A7567BA4FFA6" name="X8524A7567BA4FFA6"></a></p>
<div class="ChapSects"><a href="chap4.html#X8524A7567BA4FFA6">4 <span class="Heading">Cones</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X837F56037886A1EF">4.1 <span class="Heading">Creating cones</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E6FB9EC7DFF5403">4.1-1 ConeByInequalities</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8711F345805C5FBD">4.1-2 ConeByEqualitiesAndInequalities</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F0634207B3A4EC6">4.1-3 Cone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CD248238093A748">4.1-4 Cone</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X85EE90A37DE1E3AC">4.2 <span class="Heading">Attributes of Cones</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X80468DD7834C2EFD">4.2-1 DefiningInequalities</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X782F9A6E82AC6EA4">4.2-2 EqualitiesOfCone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X80DB81B58590FC36">4.2-3 DualCone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X825F3C9380C4E2EE">4.2-4 FacesOfCone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82C77B2A8235567E">4.2-5 Facets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X817CAD1E84C0757A">4.2-6 FVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81488179780BE42F">4.2-7 RelativeInteriorRay</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7EB6BB087BE2FFD6">4.2-8 HilbertBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8445477A83A137CB">4.2-9 HilbertBasisOfDualCone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X815F2FC4821A961D">4.2-10 LinealitySpaceGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CFFCB5C7F4C183D">4.2-11 ExternalCddCone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79B4436C8051BA08">4.2-12 ExternalNmzCone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83709F0380B097BD">4.2-13 AmbientSpaceDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7DC52F6B8490C5C4">4.2-14 LatticePointsGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B0DA052877CE7BC">4.2-15 GridGeneratedByCone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8267BFA47CD282E1">4.2-16 FactorGrid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A807C828273D4ED">4.2-17 FactorGridMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X849EFCA77CE04DF9">4.2-18 GridGeneratedByOrthogonalCone</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7DD2805F7DF61F97">4.3 <span class="Heading">Properties of Cones</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8490C19A78ED287A">4.3-1 IsRegularCone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X831EBE117FC07C40">4.3-2 IsRay</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X810956D0823BDA51">4.3-3 IsZero</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X843CBBD37FC2827B">4.4 <span class="Heading">Operations on cones</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X782E6C6684E1514D">4.4-1 FourierProjection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X80715AC378616F14">4.4-2 IntersectionOfCones</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7946B20D834F31D2">4.4-3 IntersectionOfCones</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X825A7CBD79653961">4.4-4 Contains</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81FBC83B87053C56">4.4-5 IsRelativeInteriorRay</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C52373C83B9282D">4.4-6 NonReducedInequalities</a></span>
</div></div>
</div>

<h3>4 <span class="Heading">Cones</span></h3>

<p><a id="X837F56037886A1EF" name="X837F56037886A1EF"></a></p>

<h4>4.1 <span class="Heading">Creating cones</span></h4>

<p><a id="X7E6FB9EC7DFF5403" name="X7E6FB9EC7DFF5403"></a></p>

<h5>4.1-1 ConeByInequalities</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConeByInequalities</code>( <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <code class="code">Cone</code> Object</p>

<p>The function takes a list of lists <span class="Math">[L_1, L_2, ...]</span> where each <span class="Math">L_j</span> represents an inequality and returns the cone defined by them. For example the <span class="Math">j</span>'th entry L_j = [a_{j1},a_{j2},...,a_{jn}] corresponds to the inequality \sum_{i=1}^n a_{ji}x_i \geq 0.



<p><a id="X8711F345805C5FBD" name="X8711F345805C5FBD"></a></p>

<h5>4.1-2 ConeByEqualitiesAndInequalities</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConeByEqualitiesAndInequalities</code>( <var class="Arg">Eq</var>, <var class="Arg">Ineq</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <code class="code">Cone</code> Object</p>

<p>The function takes two lists. The first list is the equalities and the second is the inequalities and returns the cone defined by them.</p>

<p><a id="X7F0634207B3A4EC6" name="X7F0634207B3A4EC6"></a></p>

<h5>4.1-3 Cone</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cone</code>( <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <code class="code">Cone</code> Object</p>

<p>The function takes a list in which every entry represents a ray in the ambient vector space and returns the cone defined by them.</p>

<p><a id="X7CD248238093A748" name="X7CD248238093A748"></a></p>

<h5>4.1-4 Cone</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cone</code>( <var class="Arg">cdd_cone</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <code class="code">Cone</code> Object</p>

<p>This function takes a cone defined in <em>CddInterface</em> and converts it to a cone in <em>NConvex</em></p>

<p><a id="X85EE90A37DE1E3AC" name="X85EE90A37DE1E3AC"></a></p>

<h4>4.2 <span class="Heading">Attributes of Cones</span></h4>

<p><a id="X80468DD7834C2EFD" name="X80468DD7834C2EFD"></a></p>

<h5>4.2-1 DefiningInequalities</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DefiningInequalities</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>Returns the list of the defining inequalities of the cone <code class="code">C</code>.</p>

<p><a id="X782F9A6E82AC6EA4" name="X782F9A6E82AC6EA4"></a></p>

<h5>4.2-2 EqualitiesOfCone</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EqualitiesOfCone</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>Returns the list of the equalities in the defining inequalities of the cone <code class="code">C</code>.</p>

<p><a id="X80DB81B58590FC36" name="X80DB81B58590FC36"></a></p>

<h5>4.2-3 DualCone</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DualCone</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a cone</p>

<p>Returns the dual cone of the cone <code class="code">C</code>.</p>

<p><a id="X825F3C9380C4E2EE" name="X825F3C9380C4E2EE"></a></p>

<h5>4.2-4 FacesOfCone</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FacesOfCone</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of cones</p>

<p>Returns the list of all faces of the cone <code class="code">C</code>.</p>

<p><a id="X82C77B2A8235567E" name="X82C77B2A8235567E"></a></p>

<h5>4.2-5 Facets</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Facets</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of cones</p>

<p>Returns the list of all facets of the cone <code class="code">C</code>.</p>

<p><a id="X817CAD1E84C0757A" name="X817CAD1E84C0757A"></a></p>

<h5>4.2-6 FVector</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FVector</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>Returns a list whose <span class="Math">i</span>'th entry is the number of faces of dimension i.



<p><a id="X81488179780BE42F" name="X81488179780BE42F"></a></p>

<h5>4.2-7 RelativeInteriorRay</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RelativeInteriorRay</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>Returns a relative interior point (or ray) in the cone <code class="code">C</code>.</p>

<p><a id="X7EB6BB087BE2FFD6" name="X7EB6BB087BE2FFD6"></a></p>

<h5>4.2-8 HilbertBasis</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HilbertBasis</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>Returns the Hilbert basis of the cone <code class="code">C</code></p>

<p><a id="X8445477A83A137CB" name="X8445477A83A137CB"></a></p>

<h5>4.2-9 HilbertBasisOfDualCone</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HilbertBasisOfDualCone</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>Returns the Hilbert basis of the dual cone of the cone <code class="code">C</code></p>

<p><a id="X815F2FC4821A961D" name="X815F2FC4821A961D"></a></p>

<h5>4.2-10 LinealitySpaceGenerators</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LinealitySpaceGenerators</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>Returns a basis of the lineality space of the cone <code class="code">C</code>.</p>

<p><a id="X7CFFCB5C7F4C183D" name="X7CFFCB5C7F4C183D"></a></p>

<h5>4.2-11 ExternalCddCone</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExternalCddCone</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a cdd object</p>

<p>Converts the cone to a cdd object. The operations of CddInterface can then be applied on this convex object.</p>

<p><a id="X79B4436C8051BA08" name="X79B4436C8051BA08"></a></p>

<h5>4.2-12 ExternalNmzCone</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExternalNmzCone</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: an normaliz object</p>

<p>Converts the cone to a normaliz object. The operations of NormalizInterface can then be applied on this convex object.</p>

<p><a id="X83709F0380B097BD" name="X83709F0380B097BD"></a></p>

<h5>4.2-13 AmbientSpaceDimension</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AmbientSpaceDimension</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: an integer</p>

<p>The dimension of the ambient space of the cone, i.e., the space that contains the cone.</p>

<p><a id="X7DC52F6B8490C5C4" name="X7DC52F6B8490C5C4"></a></p>

<h5>4.2-14 LatticePointsGenerators</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LatticePointsGenerators</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list</p>

<p>See <code class="code">LatticePointsGenerators</code> for polyhedrons. Please note that any cone is a polyhedron.</p>

<p><a id="X7B0DA052877CE7BC" name="X7B0DA052877CE7BC"></a></p>

<h5>4.2-15 GridGeneratedByCone</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GridGeneratedByCone</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a homalg module</p>

<p>Returns the homalg <span class="Math">\mathbb{Z}</span>-module that is generated by the ray generators of the cone.</p>

<p><a id="X8267BFA47CD282E1" name="X8267BFA47CD282E1"></a></p>

<h5>4.2-16 FactorGrid</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FactorGrid</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a homalg module</p>

<p>Returns the homalg <span class="Math">\mathbb{Z}</span>-module that is presented by the matrix whose raws are the ray generators of the cone.</p>

<p><a id="X7A807C828273D4ED" name="X7A807C828273D4ED"></a></p>

<h5>4.2-17 FactorGridMorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FactorGridMorphism</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a homalg morphism</p>

<p>Returns an epimorphism from a free <span class="Math">\mathbb{Z}</span>-module to <code class="code">FactorGrid(C)</code>.</p>

<p><a id="X849EFCA77CE04DF9" name="X849EFCA77CE04DF9"></a></p>

<h5>4.2-18 GridGeneratedByOrthogonalCone</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GridGeneratedByOrthogonalCone</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a homalg module</p>

<p>Returns the homalg <span class="Math">\mathbb{Z}</span>-module that is by generated the ray generators of the orthogonal cone on <code class="code">C</code>.</p>

<p><a id="X7DD2805F7DF61F97" name="X7DD2805F7DF61F97"></a></p>

<h4>4.3 <span class="Heading">Properties of Cones</span></h4>

<p><a id="X8490C19A78ED287A" name="X8490C19A78ED287A"></a></p>

<h5>4.3-1 IsRegularCone</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRegularCone</code>( <var class="Arg">C</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: true or false</p>

<p>Returns if the cone <code class="code">C</code> is regular or not.</p>

<p><a id="X831EBE117FC07C40" name="X831EBE117FC07C40"></a></p>

<h5>4.3-2 IsRay</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRay</code>( <var class="Arg">C</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: true or false</p>

<p>Returns if the cone <code class="code">C</code> is ray or not.</p>

<p><a id="X810956D0823BDA51" name="X810956D0823BDA51"></a></p>

<h5>4.3-3 IsZero</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsZero</code>( <var class="Arg">C</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: true or false</p>

<p>Returns whether the cone is the zero cone or not.</p>

<p><a id="X843CBBD37FC2827B" name="X843CBBD37FC2827B"></a></p>

<h4>4.4 <span class="Heading">Operations on cones</span></h4>

<p><a id="X782E6C6684E1514D" name="X782E6C6684E1514D"></a></p>

<h5>4.4-1 FourierProjection</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FourierProjection</code>( <var class="Arg">C</var>, <var class="Arg">m</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a cone</p>

<p>Returns the projection of the cone on the space (O, <span class="Math">x_1,...,x_{m-1}, x_{m+1},...,x_n</span> ).</p>

<p><a id="X80715AC378616F14" name="X80715AC378616F14"></a></p>

<h5>4.4-2 IntersectionOfCones</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IntersectionOfCones</code>( <var class="Arg">C1</var>, <var class="Arg">C2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a cone</p>

<p>Returns the intersection.</p>

<p><a id="X7946B20D834F31D2" name="X7946B20D834F31D2"></a></p>

<h5>4.4-3 IntersectionOfCones</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IntersectionOfCones</code>( <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a cone</p>

<p>The input is a list of cones and the output is their intersection.</p>

<p><a id="X825A7CBD79653961" name="X825A7CBD79653961"></a></p>

<h5>4.4-4 Contains</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Contains</code>( <var class="Arg">C1</var>, <var class="Arg">C2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a true or false</p>

<p>Returns if the cone <code class="code">C1</code> contains the cone <code class="code">C2</code>.</p>

<p><a id="X81FBC83B87053C56" name="X81FBC83B87053C56"></a></p>

<h5>4.4-5 IsRelativeInteriorRay</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRelativeInteriorRay</code>( <var class="Arg">L</var>, <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a true or false</p>

<p>Checks whether the input point (or ray) <code class="code">L</code> is in the relative interior of the cone <code class="code">C</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">P:= Cone( [ [ 2, 7 ], [ 0, 12 ], [ -2, 5 ] ] );</span>
<A cone in |R^2>
<span class="GAPprompt">gap></span> <span class="GAPinput">d:= DefiningInequalities( P );</span>
[ [ -7, 2 ], [ 5, 2 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Q:= ConeByInequalities( d );</span>
<A cone in |R^2>
<span class="GAPprompt">gap></span> <span class="GAPinput">P=Q;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPointed( P );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( P );</span>
[ [ -2, 5 ], [ 2, 7 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">HilbertBasis( P );</span>
[ [ -2, 5 ], [ -1, 3 ], [ 0, 1 ], [ 1, 4 ], [ 2, 7 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">HilbertBasis( Q );</span>
[ [ -2, 5 ], [ -1, 3 ], [ 0, 1 ], [ 1, 4 ], [ 2, 7 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">P_dual:= DualCone( P );</span>
<A cone in |R^2>
<span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( P_dual );</span>
[ [ -7, 2 ], [ 5, 2 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( P );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">List( Facets( P ), RayGenerators );</span>
[ [ [ -2, 5 ] ], [ [ 2, 7 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">faces := FacesOfCone( P );</span>
[ <A cone in |R^2>, <A cone in |R^2>, <A ray in |R^2>, 
 <A ray in |R^2> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RelativeInteriorRay( P );</span>
[ -2, 41 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRelativeInteriorRay( [ -2, 41 ], P );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRelativeInteriorRay( [ 2, 7 ], P );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">LinealitySpaceGenerators( P );</span>
[  ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRegularCone( P );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRay( P );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">proj_x1:= FourierProjection( P, 2 );</span>
<A cone in |R^1>
<span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( proj_x1 );</span>
[ [ -1 ], [ 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DefiningInequalities( proj_x1 );</span>
[ [ 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">R:= Cone( [ [ 4, 5 ], [ -2, 1 ] ] );</span>
<A cone in |R^2>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= IntersectionOfCones( P, R );</span>
<A cone in |R^2>
<span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( T );</span>
[ [ -2, 5 ], [ 2, 7 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= Cone( [ [-3,-4 ] ] );</span>
<A ray in |R^2>
<span class="GAPprompt">gap></span> <span class="GAPinput">I:= IntersectionOfCones( P, W );</span>
<A cone in |R^2>
<span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( I );</span>
[  ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Contains( P, I );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Contains( W, I );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Contains( P, R );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Contains( R, P );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">cdd_cone:= ExternalCddCone( P );</span>
< Polyhedron given by its V-representation >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( cdd_cone );</span>
V-representation 
begin 
3 X 3  rational
               
   0   2   7 
   0   0  12 
   0  -2   5 
end
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_Dimension( cdd_cone );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">H:= Cdd_H_Rep( cdd_cone );</span>
< Polyhedron given by its H-representation >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( H );</span>
H-representation 
begin 
   2 X 3  rational
               
   0   5   2 
   0  -7   2 
end
<span class="GAPprompt">gap></span> <span class="GAPinput">P:= Cone( [ [ 1, 1, -3 ], [ -1, -1, 3 ], [ 1, 2, 1 ], [ 2, 1, 2 ] ] );</span>
< A cone in |R^3>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPointed( P );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( P );</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRegularCone( P );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">P;</span>
< A cone in |R^3 of dimension 3 with 4 ray generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( P );</span>
[ [ -1, -1, 3 ], [ 1, 1, -3 ], [ 1, 2, 1 ], [ 2, 1, 2 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">d:= DefiningInequalities( P );</span>
[ [ -5, 8, 1 ], [ 7, -4, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">facets:= Facets( P );</span>
[ <A cone in |R^3>, <A cone in |R^3> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">faces := FacesOfCone( P );</span>
[ <A cone in |R^3>, <A cone in |R^3>, <A cone in |R^3>, 
 <A cone in |R^3>, <A cone in |R^3> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FVector( P );</span>
[ 1, 2, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( faces, Dimension );</span>
[ 0, 3, 2, 1, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">L_using_4ti2 := [ [ [ 0, 0, 0 ] ], [ [ -2, -1, 10 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput">[ 0, 0, 1 ], [ 2, 1, 2 ] ],  [ [ 1, 1, -3 ] ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L_using_Normaliz := [ [ [ 0, 0, 0 ] ], [ [ -1, 0, 7 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput">[ 0, 0, 1 ], [ 1, 0, 5 ] ], [ [ 1, 1, -3 ] ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := LatticePointsGenerators( P );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L = L_using_4ti2 or L = L_using_Normaliz;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">DualCone( P );</span>
< A cone in |R^3>
<span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( DualCone( P ) );</span>
[ [ -5, 8, 1 ], [ 7, -4, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Q_x1x3:= FourierProjection(P, 2 );</span>
<A cone in |R^2>
<span class="GAPprompt">gap></span> <span class="GAPinput">RayGenerators( Q_x1x3 );</span>
[ [ -1, 3 ], [ 1, -3 ], [ 1, 1 ] ]
</pre></div>

<p><a id="X7C52373C83B9282D" name="X7C52373C83B9282D"></a></p>

<h5>4.4-6 NonReducedInequalities</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NonReducedInequalities</code>( <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a list</p>

<p>It returns a list of inequalities that define the cone.</p>


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