Polytopes
Creating polytopes a Polytope Object The operation takes a list L of lists [L_1, L_2, ...] where each L_j represents an inequality and returns the polytope defined by them (if they define a polytope). For example the j'th entry L_j = [c_j,a_{j1},a_{j2},...,a_{jn}] corresponds to the inequality c_j+\sum_{i=1}^n a_{ji}x_i \geq 0. a Polytope Object The operation takes the list of the vertices and returns the polytope defined by them.
Attributes a CddPolyhedron Converts the polytope to a CddPolyhedron. The operations of CddInterface can then be applied on this polyhedron. a List The operation returns the list of integer points inside the polytope. a List The operation returns the interior lattice points inside the polytope. a list of lists The operation returns the vertices of the polytope a list of lists The same output as VerticesOfPolytope. a list of lists The operation returns the defining inequalities of the polytope. I.e., a list of lists [L_1, L_2, ...] where each L_j=[c_j,a_{j1},a_{j2},...,a_{jn}] represents the inequality c_j+\sum_{i=1}^n a_{ji}x_i \geq 0. If L and -L occur in the output then L is called a defining-equality of the polytope. a list of lists The operation returns the defining-equalities of the polytope. a list of lists The operation returns the list of the inequalities of the facets. Each defining inequality that is not defining-equality of the polytope is a facet inequality. a list of lists The operation returns list of lists L. The entries of each L_j in L consists of 0's or 1's. For instance, if L_j=[1,0,0,1,0,1], then The polytope has 6 vertices and the vertices of the j'th facet are \{V_1,V_4,V_6\}. a fan The operation returns the normal fan of the given polytope. a fan The operation returns the face fan of the given polytope. Remember that the face fan of a polytope is isomorphic to the normal fan of its polar polytope. a cone If the ambient space of the polytope is \mathrm{R}^n, then the output is a cone in \mathrm{R}^{n+1}. The defining rays of the cone are {[a_{j1},a_{j2},...,a_{jn},1]}_j such that V_j=[a_{j1},a_{j2},...,a_{jn}] is a vertex in the polytope. a Polytope The operation returns the polar polytope of the given polytope. a Polytope The operation returns the dual polytope of the given polytope.
Properties a true or false Returns whether the polytope empty or not a true or false Returns whether the polytope is lattice polytope or not. a true or false Returns whether the polytope is very ample or not. a true or false Returns whether the polytope is normal or not. a true or false Returns whether the polytope is simplicial or not. a true or false Returns whether the polytope is simplex polytope or not. a true or false Returns whether the polytope is simple or not. a true or false Returns whether the polytope is reflexive or not, i.e., if its dual polytope is lattice polytope. a true or false returns whether the polytope is Fano or not. Fano polytope is a full dimensional lattice polytope whose vertices are primitive elements in the containing lattice, i.e., each vertex is not a positive integer multiple of any other lattice element. a true or false returns whether the polytope is canonical Fano or not. A canonical Fano polytope is a full dimensional lattice polytope whose relative interior contains only one lattice point, namely the origin. a true or false returns whether the polytope is terminal Fano or not. A terminal Fano polytope is a full dimensional lattice polytope whose lattice points are its vertices and the origin. a true or false Returns whether the polytope is smooth fano polytope or not, i.e, if the vertices in each facet form a basis for the containing lattice or not. polytope.
Operations on polytopes a polytope The output is Minkowski sum of the input polytopes. a polytope The output is Minkowski sum of the input polytope with itself n times. a polytope The output is the intersection of the input polytopes. a list Returns a random interior point in the polytope. true or false Checks if the given point is interior point of the polytope. <#Include Label="example2">