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Journal of Convex Analysis 26 (2019), No. 3, 1001--1019
Copyright Heldermann Verlag 2019

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

Marek Niezgoda
Dept. of Applied Mathematics and Computer Science, University of Life Sciences, 20-950 Lublin, Poland


\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 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.

Keywords: Convex function, eigenvalues, singular value, G-invariant function, G-majorization, Eaton triple, normal decomposition system, normal map, psi-uniformly convex function.

MSC: 15A30, 15A21; 26B25, 06F20

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