Normal fans of polyhedral convex sets
type: Item
instance of: scholarly article
title: Normal fans of polyhedral convex sets
zbMATH Open document ID: 1144.52008
author: Shu Lu
author: Stephen M. Robinson
published in: Set-Valued Analysis
publication date: 2008-06-25T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 52A99
Mathematics Subject Classification ID: 52B99
Mathematics Subject Classification ID: 90C31
Mathematics Subject Classification ID: 52A41
zbMATH DE Number: 5292862
zbMATH Keywords: normal cone
zbMATH Keywords: normal fan
zbMATH Keywords: convex polyhedra
zbMATH Keywords: face
zbMATH Keywords: critical cone
zbMATH Keywords: right hand side perturbation
MaRDI profile type: Publication
full work available at URL: s11228-008-0077-9
OpenAlex ID: W1548524765
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DOI: 10.1007/S11228-008-0077-9
description: scientific article; zbMATH DE number 5292862