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4.22 cone

In order to use convex objects in Singular, Singular uses gfanlib, a C++ library for convex geometry by Anders N. Jensen. Please check the readme file for installation instructions. The library gfan_lib provides this C++ library and an interface to it. Some functions require polymake: polymake_lib provides an interface to it.

In the finite dimensional real vector space R^n, a convex rational polyhedral cone, in short "cone", is the convex set generated by finitely many half-lines, which in turn are generated by rational, and hence integer, points. It may or may not contain whole subspace of R^n (e.g. entire lines). The biggest subspace contained in a cone is called "lineality space". Modulo its lineality space, each cone is generated by a distinct minimal set of half lines, which are referred to as "rays". Alternatively, a cone can be represented as a set of points satisfying a system homogeneous rational, and hence integer, linear inequalities and equations. These two characterizations of cones are the two main ways of defining cones in Singular (see coneViaPoints, see coneViaInequalities).

cone c;                             // ambient dim 0, no equations,
                                    // no inequalities
cone c = 17;                        // ambient dim 17, no equations,
                                    // no inequalities

See gfan_lib.