An Eaton triple is an algebraic system related to a decomposition statement for vectors of an inner product space
V
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)
(
V
,
G
,
D
)
and for a function
F\colon V \to \mathbb{R}
F
:
V
→
R
, invariant with respect to the group
G
G
acting on
V
V
, we study the problem of extending convexity of
F
F
from the convex cone
D \subset V
D
⊂
V
to the space
V
V
. In our approach we reduce the problem from E-system
(V,G,D)
(
V
,
G
,
D
)
to its subsystem
(W,H,E)
(
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
ψ
-uniform convex functions and
c
c
-strongly convex functions. Finally, applications are given for matrix spaces endowed with the structure of Eaton triple.