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<h1>NConvex</h1>
<h2>A Gap package to perform polyhedral computations</h2>
<p>
2024 .12 -01 </p>
<p>
28 December 2024
</p>
</div>
<p><b>
Kamal Saleh
</b>
<br />Email: <span class ="URL" ><a href="mailto:kamal.saleh@uni-siegen.de" >kamal.saleh@uni-siegen.de</a></span>
<br />Homepage: <span class ="URL" ><a href="https://github.com/kamalsaleh ">https://github.com/kamalsaleh</a></span>
<br />Address: <br />Department Mathematik<br /> Universität Siegen<br /> Walter-Flex-Straße 3 <br /> 57068 Siegen<br /> Germany<br />
</p><p><b>
Sebastian Gutsche
</b>
<br />Email: <span class ="URL" ><a href="mailto:gutsche@mathematik.uni-siegen.de" >gutsche@mathematik.uni-siegen.de</a></span>
<br />Homepage: <span class ="URL" ><a href="https://sebasguts.github.io/ ">https://sebasguts.github.io/</a></span>
<br />Address: <br />Department Mathematik<br /> Universität Siegen<br /> Walter-Flex-Straße 3 <br /> 57068 Siegen<br /> Germany<br />
</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8" ></a></p>
<div class ="contents" >
<h3>Contents<a id="contents" name="contents" ></a></h3>
<div class ="ContChap" ><a href="chap1_mj.html#X7DFB63A97E67C0A1" >1 <span class ="Heading" >Introduction</span></a>
<div class ="ContSect" ><span class ="tocline" ><span class ="nocss" > </span><a href="chap1_mj.html#X8360C04082558A12" >1 .1 <span class ="Heading" >Installation</span></a>
</span>
</div>
<div class ="ContSect" ><span class ="tocline" ><span class ="nocss" > </span><a href="chap1_mj.html#X85A08CF187A6D986" >1 .2 <span class ="Heading" >Requirements</span></a>
</span>
</div>
</div>
<div class ="ContChap" ><a href="chap2_mj.html#X87F1120883F5B4D0" >2 <span class ="Heading" >Functionality</span></a>
</div>
<div class ="ContChap" ><a href="chap3_mj.html#X8359268B7FDA6AEC" >3 <span class ="Heading" >Convex objects</span></a>
<div class ="ContSect" ><span class ="tocline" ><span class ="nocss" > </span><a href="chap3_mj.html#X7C701DBF7BAE649A" >3 .1 <span class ="Heading" >Attributes</span></a>
</span>
<div class ="ContSSBlock" >
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap3_mj.html#X7D87C111848F25C5" >3 .1 -1 AmbientSpaceDimension</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap3_mj.html#X79D9BB0F7BB3B411" >3 .1 -2 Dimension</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap3_mj.html#X7E660B6A7B6D0A9D" >3 .1 -3 IsFullDimensional</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap3_mj.html#X8401C12C7937C521" >3 .1 -4 InteriorPoint</a></span>
</div></div>
</div>
<div class ="ContChap" ><a href="chap4_mj.html#X8524A7567BA4FFA6" >4 <span class ="Heading" >Cones</span></a>
<div class ="ContSect" ><span class ="tocline" ><span class ="nocss" > </span><a href="chap4_mj.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_mj.html#X7E6FB9EC7DFF5403" >4 .1 -1 ConeByInequalities</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X8711F345805C5FBD" >4 .1 -2 ConeByEqualitiesAndInequalities</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X7F0634207B3A4EC6" >4 .1 -3 Cone</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X7CD248238093A748" >4 .1 -4 Cone</a></span>
</div></div>
<div class ="ContSect" ><span class ="tocline" ><span class ="nocss" > </span><a href="chap4_mj.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_mj.html#X80468DD7834C2EFD" >4 .2 -1 DefiningInequalities</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X782F9A6E82AC6EA4" >4 .2 -2 EqualitiesOfCone</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X80DB81B58590FC36" >4 .2 -3 DualCone</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X825F3C9380C4E2EE" >4 .2 -4 FacesOfCone</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X82C77B2A8235567E" >4 .2 -5 Facets</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X817CAD1E84C0757A" >4 .2 -6 FVector</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X81488179780BE42F" >4 .2 -7 RelativeInteriorRay</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X7EB6BB087BE2FFD6" >4 .2 -8 HilbertBasis</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X8445477A83A137CB" >4 .2 -9 HilbertBasisOfDualCone</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X815F2FC4821A961D" >4 .2 -10 LinealitySpaceGenerators</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X7CFFCB5C7F4C183D" >4 .2 -11 ExternalCddCone</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X79B4436C8051BA08" >4 .2 -12 ExternalNmzCone</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X83709F0380B097BD" >4 .2 -13 AmbientSpaceDimension</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X7DC52F6B8490C5C4" >4 .2 -14 LatticePointsGenerators</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X7B0DA052877CE7BC" >4 .2 -15 GridGeneratedByCone</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X8267BFA47CD282E1" >4 .2 -16 FactorGrid</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X7A807C828273D4ED" >4 .2 -17 FactorGridMorphism</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X849EFCA77CE04DF9" >4 .2 -18 GridGeneratedByOrthogonalCone</a></span>
</div></div>
<div class ="ContSect" ><span class ="tocline" ><span class ="nocss" > </span><a href="chap4_mj.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_mj.html#X8490C19A78ED287A" >4 .3 -1 IsRegularCone</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X831EBE117FC07C40" >4 .3 -2 IsRay</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X810956D0823BDA51" >4 .3 -3 IsZero</a></span>
</div></div>
<div class ="ContSect" ><span class ="tocline" ><span class ="nocss" > </span><a href="chap4_mj.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_mj.html#X782E6C6684E1514D" >4 .4 -1 FourierProjection</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X80715AC378616F14" >4 .4 -2 IntersectionOfCones</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X7946B20D834F31D2" >4 .4 -3 IntersectionOfCones</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X825A7CBD79653961" >4 .4 -4 Contains</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X81FBC83B87053C56" >4 .4 -5 IsRelativeInteriorRay</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap4_mj.html#X7C52373C83B9282D" >4 .4 -6 NonReducedInequalities</a></span>
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</div>
<div class ="ContChap" ><a href="chap5_mj.html#X7EB0337A86DDD2F1" >5 <span class ="Heading" >Fans</span></a>
<div class ="ContSect" ><span class ="tocline" ><span class ="nocss" > </span><a href="chap5_mj.html#X86EC0F0A78ECBC10" >5 .1 <span class ="Heading" >Constructors</span></a>
</span>
<div class ="ContSSBlock" >
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X7F5B09727CD79D17" >5 .1 -1 Fan</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X7FB87B4983087091" >5 .1 -2 Fan</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X7A1FD0BD845E3588" >5 .1 -3 Fan</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X7D5DE7F08555C8E8" >5 .1 -4 FansFromTriangulation</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X853CC6BB854AA139" >5 .1 -5 FanFromTriangulation</a></span>
</div></div>
<div class ="ContSect" ><span class ="tocline" ><span class ="nocss" > </span><a href="chap5_mj.html#X7C701DBF7BAE649A" >5 .2 <span class ="Heading" >Attributes</span></a>
</span>
<div class ="ContSSBlock" >
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X789156E3828930D2" >5 .2 -1 RayGenerators</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X86CB98CC80E9C36E" >5 .2 -2 GivenRayGenerators</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X7B0FAF0A8246F5AE" >5 .2 -3 RaysInMaximalCones</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X7D586B547A8DB803" >5 .2 -4 MaximalCones</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X7E5274F987E8FFC5" >5 .2 -5 FVector</a></span>
</div></div>
<div class ="ContSect" ><span class ="tocline" ><span class ="nocss" > </span><a href="chap5_mj.html#X871597447BB998A1" >5 .3 <span class ="Heading" >Properties</span></a>
</span>
<div class ="ContSSBlock" >
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X81189D5287FFBEDF" >5 .3 -1 IsWellDefinedFan</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X87963F4C84C9EA21" >5 .3 -2 IsComplete</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X7BE6F87D7BB9A6CD" >5 .3 -3 IsPointed</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X7C694ECC81F4DECF" >5 .3 -4 IsSmooth</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X78A031D8848E0FF8" >5 .3 -5 IsSimplicial</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X85C533997DC6B46A" >5 .3 -6 IsNormalFan</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X7A2AE62C86CE72B8" >5 .3 -7 IsRegularFan</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap5_mj.html#X81D74C107DBB7C21" >5 .3 -8 IsFanoFan</a></span>
</div></div>
<div class ="ContSect" ><span class ="tocline" ><span class ="nocss" > </span><a href="chap5_mj.html#X86BC038782EFD451" >5 .4 <span class ="Heading" >Operations on fans</span></a>
</span>
</div>
</div>
<div class ="ContChap" ><a href="chap6_mj.html#X85633D87813F12B0" >6 <span class ="Heading" >Polyhedrons</span></a>
<div class ="ContSect" ><span class ="tocline" ><span class ="nocss" > </span><a href="chap6_mj.html#X7C1A095983C57A83" >6 .1 <span class ="Heading" >Creating polyhedron</span></a>
</span>
<div class ="ContSSBlock" >
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X83EF304B85ED4C5B" >6 .1 -1 PolyhedronByInequalities</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X842F90887840D3B9" >6 .1 -2 Polyhedron</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X861DE99B8007A4C4" >6 .1 -3 Polyhedron</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X78BCB2888114F56F" >6 .1 -4 Polyhedron</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X7A8ECB9B8542EDFF" >6 .1 -5 Polyhedron</a></span>
</div></div>
<div class ="ContSect" ><span class ="tocline" ><span class ="nocss" > </span><a href="chap6_mj.html#X7C701DBF7BAE649A" >6 .2 <span class ="Heading" >Attributes</span></a>
</span>
<div class ="ContSSBlock" >
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X7ECCE0E480D88952" >6 .2 -1 ExternalCddPolyhedron</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X7C7CC5497FC52B67" >6 .2 -2 ExternalNmzPolyhedron</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X842632AB7A09A820" >6 .2 -3 DefiningInequalities</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X87375C8586090666" >6 .2 -4 MainRatPolytope</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X7A16114A7FFB3309" >6 .2 -5 MainPolytope</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X7A3FB56D7E895535" >6 .2 -6 VerticesOfMainRatPolytope</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X8697AC27862160FE" >6 .2 -7 VerticesOfMainPolytope</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X7C248476851AEB4A" >6 .2 -8 TailCone</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X7DB9524D7C1A5361" >6 .2 -9 RayGeneratorsOfTailCone</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X79351A698787C5CA" >6 .2 -10 LatticePointsGenerators</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X7B57BEAF78559D12" >6 .2 -11 BasisOfLinealitySpace</a></span>
<span class ="ContSS" ><br /><span class ="nocss" > </span><a href="chap6_mj.html#X81435DD28524EEE8" >6 .2 -12 FVector</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X871597447BB998A1">6 .3 <span class="Heading">Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7B25E29378AF4489">6 .3 -1 IsBounded</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7B8DDCE685798D9C">6 .3 -2 IsPointed</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X819F85537D5D9D15">6 .4 <span class="Heading">Solving Linear programs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X879BD74A7F6327B2">6 .4 -1 SolveLinearProgram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7F4FE0C883B8AA50">6 .4 -2 SolveLinearProgram</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X84237872798DB501">6 .5 <span class="Heading">ZSolve</span></a>
</span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X81CA403280755A9B">6 .5 -1 SolveEqualitiesAndInequalitiesOverIntergers</a></span>
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<div class="ContChap"><a href="chap7_mj.html#X7B933C4686727183">7 <span class="Heading">Polytopes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X801887DC7E5F6AB7">7 .1 <span class="Heading">Creating polytopes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X83FB00B383A19F03">7 .1 -1 PolytopeByInequalities</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X81DC4EE8849B08DA">7 .1 -2 Polytope</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7C701DBF7BAE649A">7 .2 <span class="Heading">Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7D8727917B72396A">7 .2 -1 ExternalCddPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X83B1CBC67B34BE04">7 .2 -2 LatticePoints</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7FB9E36C7B6C616F">7 .2 -3 RelativeInteriorLatticePoints</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X816ACBEF786C23F2">7 .2 -4 VerticesOfPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X805FCCD583DACF8F">7 .2 -5 Vertices</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7B39989A7F321147">7 .2 -6 DefiningInequalities</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7DAF68C08043105D">7 .2 -7 EqualitiesOfPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7FDABF3D84E67F2C">7 .2 -8 FacetInequalities</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7EC0B0818106EA50">7 .2 -9 VerticesInFacets</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7DE7E52D800A7258">7 .2 -10 NormalFan</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8453091478A4C0F1">7 .2 -11 FaceFan</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X85EC550B7FCAECB5">7 .2 -12 AffineCone</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7ACC98FD78ACDC57">7 .2 -13 PolarPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8349F8DA81418C35">7 .2 -14 DualPolytope</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X871597447BB998A1">7 .3 <span class="Heading">Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X863F64947BDFC654">7 .3 -1 IsEmpty</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X85181A5986107254">7 .3 -2 IsLatticePolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7918DAE87B159619">7 .3 -3 IsVeryAmple</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X851AD9E684CA7434">7 .3 -4 IsNormalPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7F39F2BF835AA337">7 .3 -5 IsSimplicial</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X797F6A0778C70DD0">7 .3 -6 IsSimplexPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X815A80E785E52474">7 .3 -7 IsSimplePolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X85BE2236780D0957">7 .3 -8 IsReflexive</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X85DEC39081FD04E9">7 .3 -9 IsFanoPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X780D5BDE7C177166">7 .3 -10 IsCanonicalFanoPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X78B2B9747E9A6FEA">7 .3 -11 IsTerminalFanoPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7946CEA886A48F96">7 .3 -12 IsSmoothFanoPolytope</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7B71F2A2834E84CE">7 .4 <span class="Heading">Operations on polytopes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7F9303847F938C87"><code>7 .4 -1 \+</code></a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7AEF19987E29612E"><code>7 .4 -2 \*</code></a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8079AF527BA681D1">7 .4 -3 IntersectionOfPolytopes</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7F2AE1197AAFF273">7 .4 -4 RandomInteriorPoint</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7E36517B808D2E44">7 .4 -5 IsInteriorPoint</a></span>
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