Extreme positive operators on minimal and almost minimal cones

Abstract

Fiedler and Pták called a cone minimal if it is n-dimensional and has n+1 extremal rays. We call a cone almost minimal if it is n-dimensional and has n+2 extremal rays. Duality properties stemming from the use of Gale pairs lead to a general technique for identifying the extreme cone-preserving (positive) operators between polyhedral cones. This technique is most effective for cones with dimension not much smaller than the number of their extreme rays. In particular, the Fiedler-Pták characterization of extreme positive operators between minimal cones is extended to the following cases: (i) operators from a minimal cone to an arbitrary polyhedral cone, (ii) operators from an almost minimal cone to a minimal cone.