Normal fans of polyhedral convex sets
The normal cone of a face \(F\) of a polyhedral convex set is given by all linear functionals that reach its maximum at face \(F\). The normal fan of a polyhedral convex set in \(\mathbb R^{n}\) is the collection of all its normal cones. The structure of the normal fan reflects the geometry of that polyhedron. For example two polyhedral convex sets with the same normal fan have the same face lattice. These notions have become important not only for convex analysis but also in other areas of mathematics such as combinatorics and algebra (the authors mention a few but the use of this concept is more prevalent, for example in computer algebra, the Gröbner fan or state fan of a polynomial ideal are normal fans of polytopes associated to Gröbner bases). This paper reviews and studies properties about the normal fan. In particular, it investigates situations in which the normal fan of a polyhedral convex set refines, or is a subfan of, that of another set. It then applies these techniques in several examples. One of these concerns the face structure and normal manifold of the critical cone of a polyhedral convex set associated with a point in \(\mathbb R^{n}\). Another application concerns how perturbation of the right hand side of the linear constraints defining such a set affects the normal fan and the face structure.
scientific article; zbMATH DE number 5292862