On Convexity and ψ-Uniform Convexity of G-Invariant Functions on an Eaton Triple

An Eaton triple is an algebraic system related to a decomposition statement for vectors of an inner product space $V$ and to some special inner product inequality connected with this decomposition. The Spectral Decomposition for the space of Hermitian matrices associated with Fan-Theobald's trace inequality is a typical example of such a situation. In this paper, for a given Eaton triple $(V,G,D)$ and for a function $F\colon V \to \R$, invariant with respect to the group $G$ acting on $V$, we study the problem of extending convexity of $F$ from the convex cone $D \subset V$ to the space $V$. In our approach we reduce the problem from E-system $(V,G,D)$ to its subsystem $(W,H,E)$. Thus we obtain some results related to theorems due to J.\,von Neumann, C.\,Davis, A.\,S.\,Lewis and T.-Y.\,Tam et al. Analogous problems are discussed for $\psi$-uniform convex functions and $c$-strongly convex functions. Finally, applications are given for matrix spaces endowed with the structure of Eaton triple.