Normal Fans of Polyhedral Convex Sets Structures and Connections

The normal fan of a polyhedral convex set in Rn is the collection of its normal cones. The structure of the normal fan reflects the geometry of that set. 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 Rn. Another concerns how perturbation of the right hand side of the linear constraints defining such a set affects the normal fan and the face structure. The research reported here was sponsored in part by the National Science Foundation under Grant DMS0305930, in part by the Air Force Research Laboratory under agreement numbers FA9550-04-1-0192 and FA9550-07-1-0389, and in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under Grant DAAD19-01-1-0502. The U. S. Government has certain rights in this material, and is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the sponsoring agencies or the U. S. Government. Shu Lu Department of Statistics and Operations Research 210 Smith Building, CB# 3260 The University of North Carolina at Chapel Hill Chapel Hill, NC 27599-3260, USA Tel: +1-919-962-1048 Fax: +1-919-962-0391 E-mail: [email protected] Stephen M. Robinson Department of Industrial and Systems Engineering University of Wisconsin-Madison 1513 University Ave Rm 3015 Madison, WI 53706-1539, USA Tel: +1-608-263-6862 Fax: +1-608-262-8454 E-mail: [email protected]