
Journal of Convex Analysis 26 (2019), No. 3, 10011019 Copyright Heldermann Verlag 2019 On Convexity and ψUniform Convexity of GInvariant Functions on an Eaton Triple Marek Niezgoda Dept. of Applied Mathematics and Computer Science, University of Life Sciences, 20950 Lublin, Poland marek.niezgoda@up.lublin.pl [Abstractpdf] \newcommand{\R}{\mathbb R} 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 FanTheobald'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 Esystem $(V,G,D)$ 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. Keywords: Convex function, eigenvalues, singular value, Ginvariant function, Gmajorization, Eaton triple, normal decomposition system, normal map, psiuniformly convex function. MSC: 15A30, 15A21; 26B25, 06F20 [ Fulltextpdf (138 KB)] for subscribers only. 