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Cell[TextData[{
StyleBox["cddmathlink",
FontColor->RGBColor[0.0557107, 0.137819, 0.517113]],
"\nConvex Hull and Vertex Enumeration by ",
StyleBox["MathLink",
FontSlant->"Italic",
FontColor->RGBColor[0.0146487, 0.461387, 0.0967727]],
" to ",
StyleBox["cddlib",
FontColor->RGBColor[0.517113, 0.0273594, 0.0273594]],
"\nby Komei Fukuda\nApril 17, 2001"
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Cell[CellGroupData[{
Cell["Connecting cddmathlink", "Subsection",
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Cell[CellGroupData[{
Cell[TextData[{
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In this example, the name of the directory is ",
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FontFamily->"Courier",
FontWeight->"Bold"]
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Cell["\<\
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\("AllVertices[m,d+1,A] generates all extreme points (vertices) and \
extreme rays of the convex polyhedron in R^(d+1) given as the solution set to \
an inequality system A x >= 0 where A is an m*(d+1) matrix and \
x=(1,x1,...,xd). The output is {{extlist, linearity}, ecdlist} where extlist \
is the extreme point list and ecdlist is the incidence list. Each vertex \
(ray) has the first component 1 (0). If the convex polyhedron is nonempty \
and has no vertices, extlist is a (nonunique) set of generators of the \
polyhedron where those generators in the linearity list are considered as \
linearity space (of points satisfying A (0, x1, x2, ...., xd) = 0) \
generators."\)], "Print",
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Let's try this function with a 3-dimenstional cube defined by 6 \
inequalities (facets);
x1 >= 0, x2 >=0, x3 >= 0, 1 - x1 >= 0, 1 - x2 >= 0 and 1 - x3 >= 0. We \
write these six inequalities as A x >= 0 and x=(1, x1, x2, x3).\
\>", \
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(vertices) and extreme rays of the convex polyhedron in R^(d+1) given as the \
solution set to an inequality system A x >= 0 where A is an m*(d+1) matrix \
and x=(1,x1,...,xd). The output is {{extlist, linearity}, ecdlist, eadlist, \
icdlist, iadlist} where extlist, ecdlist, eadlist are the extreme point list, \
the incidence list, the adjacency list (of extreme points and rays), and \
icdlist, iadlist are the incidence list, the adjacency list (of \
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convex polyhedron is nonempty and has no vertices, extlist is a (nonunique) \
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polyhedron in R^(d+1) generated by points and rays given in the rows of an \
n*(d+1) matrix G. Each point (ray) must have 1 (0) in the first coordinate. \
The output is {{faclist, equalities}, icdlist} where faclist is the facet \
list and icdlist is the incidence list. If the convex polyhedron is not \
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polyhedron where those inequalities in the equalities list are considered as \
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We have computed all the vertices of a 3-cube. Let's try the \
reverse operation. First check the size of the list of vertics. It should \
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\>", "Text",
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Cell["\<\
If you want to compute an inequality description of the \
one-dimensional cone in R^3 with a vertex at origin and containing the \
direction (1,1,1), you must set up the input (generator) data as:\
\>", "Text",\
ImageRegion->{{0, 1}, {0, 1}}],
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Since the equalities list contains 3 and 4, of the four output \
inequalities , the third and the forth must be considered as equalities. It \
is important to note that this cone can have infinitely many different \
minimal inequality descriptions, since it is not full-dimensional.\
\>", \
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Let's compute the convex hull of 0/1 points in R^d. First generate \
0/1 points. Below each point with the first component 0 is considered as a \
direction that must be included in the convex hull.\
\>", "Text",
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