In this paper, we consider the set $\mathcal{S} = a(e^X K e^Y)$ where $a(g)$ is the abelian part in the Cartan decomposition of $g$. This is exactly the support of the measure intervening in the product formula for the spherical functions on symmetric spaces of noncompact type. We give a simple description of that support in the case of $\SL(3,\mathbf{F})$ where $\mathbf{F} = \mathbf{R}$, $\mathbf{C}$ or $\mathbf{H}$. In particular, we show that $\mathcal{S}$ is convex. We also give an application of our result to the description of singular values of a product of two arbitrary matrices with prescribed singular values.

Given a matrix $A$, let $\mathcal{O}(A)$ denote the orbit of $A$ under a certain group action such as \begin{enumerate}[(4)] \item[(1)] $U(m) \otimes U(n)$ acting on $m \times n$ complex matrices $A$ by $(U,V)*A = UAV^t$, \item[(2)] $O(m) \otimes O(n)$ or $\SO(m) \otimes \SO(n)$ acting on $m \times n$ real matrices $A$ by $(U,V)*A = UAV^t$, \item[(3)] $U(n)$ acting on $n \times n$ complex symmetric or skew-symmetric matrices $A$ by $U*A = UAU^t$, \item[(4)] $O(n)$ or $\SO(n)$ acting on $n \times n$ real symmetric or skew-symmetric matrices $A$ by $U*A = UAU^t$. \end{enumerate} Denote by $$ \mathcal{O}(A_1,\dots,A_k) = \{X_1 + \cdots + X_k : X_i \in \mathcal{O}(A_i), i = 1,\dots,k\} $$ the joint orbit of the matrices $A_1,\dots,A_k$. We study the set of diagonals or partial diagonals of matrices in $\mathcal{O}(A_1,\dots,A_k)$, {\it i.e.}, the set of vectors $(d_1,\dots,d_r)$ whose entries lie in the $(1,j_1),\dots,(r,j_r)$ positions of a matrix in $\mathcal{O}(A_1, \dots,A_k)$ for some distinct column indices $j_1,\dots,j_r$. In many cases, complete description of these sets is given in terms of the inequalities involving the singular values of $A_1,\dots,A_k$. We also characterize those extreme matrices for which the equality cases hold. Furthermore, some convexity properties of the joint orbits are considered. These extend many classical results on matrix inequalities, and answer some questions by Miranda. Related results on the joint orbit $\mathcal{O}(A_1,\dots,A_k)$ of complex Hermitian matrices under the action of unitary similarities are also discussed.

A homogeneous real polynomial $p$ is {\em hyperbolic} with respect to a given vector $d$ if the univariate polynomial $t \mapsto p(x-td)$ has all real roots for all vectors $x$. Motivated by partial differential equations, G{\aa}rding proved in 1951 that the largest such root is a convex function of $x$, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize G{\aa}rding's result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems over related cones.