In this paper, we consider the set
$\text{S}=a\left( {{e}^{X}}K{{e}^{Y}} \right)$
where
$a\left( g \right)$
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
$\text{SL}\left( 3,\,\mathbf{F} \right)\,\text{where}\,\mathbf{F}\,=\,\mathbf{R},\,\mathbf{C}\,\text{or}\,\mathbf{H}$
. In particular, we show that
$\text{S}$
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.