G-majorization, group-induced cone orderings, and reflection groups (Q911685)

Let V be a finite dimensional real inner product space and G be a finite subgroup of the orthogonal group \({\mathcal O}(V)\). For x, \(y\in V\) x is G- majorized by y, i.e. \(x\prec y\) if and only if the convex hull of the set \(\{\) gy\(|\) \(g\in G\}\) contains x. The author points out that the following statements are equivalent: (a) there is a closed convex cone \({\mathcal T}\) in V such that for any x, \(y\in {\mathcal T}^ x \)is G-majorized by y if and only if (x,u)\(\leq (y,u)\) for all \(u\in {\mathcal T};\) (b) G is a reflection group generated by a finite number of reflections.