Generalized Schur-concave functions and Eaton triples
نویسنده
چکیده
Motivated by Schur-concavity, we introduce the notion of G-concavity where G is a closed subgroup of the orthogonal group O(V ) on a finite dimensional real inner product space V . The triple (V,G, F ) is an Eaton triple if F ⊂ V is a nonempty closed convex cone such that (A1) Gx ∩ F is nonempty for each x ∈ V . (A2) maxg∈G(x, gy) = (x, gy) for all x, y ∈ F . If W := spanF and H := {g|W : g ∈ G, gW = W} ⊂ O(W ), and (W,H, F ) is an Eaton triple, then (W,H, F ) is called a reduced triple of the Eaton triple (V, G, F ). In this event, a characterization of the G-concavity in terms of H-concavity is obtained. Some differential characterizations of G-concavity are then given. The results are applied to Lie groups. Various matrix examples are given. 2000 Mathematics Subject Classification. Primary 15A45
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