G-Majorization, group-induced cone orderings, and reflection groups

Abstract A vector y is G-majorized by a vector x if y is an element of the convex hull of the orbit of x under the action of a group G. It is known that if G is a finite reflection group, then G-majorization is equivalent to a group-induced cone ordering. In this paper it is established that if for a finite subgroup G of the orthogonal group G-majorization is equivalent to a group-induced cone ordering, then G must be a reflection group.