  We study certain pairs of subspaces $V$ and $W$ of $\mathbb{C}^n$ we call
supports that consist of eigenspaces of the eigenvalues $\pm\|M\|$ of a minimal
hermitian matrix $M$ ($\|M\|\leq \|M+D\|$ for all real diagonals $D$). For any
pair of orthogonal subspaces we define a non negative invariant $\delta$ called
the adequacy to measure how close they are to form a support and to detect one.
This function $\delta$ is the minimum of another map $F$ defined in a product
of spheres of hermitian matrices. We study the gradient, Hessian and critical
points of $F$ in order to approximate $\delta$. These results allow us to prove
that the set of supports has interior points in the space of flag manifolds.
