  We investigate convexity properties of the set of eigenvalue tuples of
$n\times n$ real symmetric matrices, whose all $k\times k$ (where $k\leq n$ is
fixed) minors are positive semidefinite. It is proven that the set
$\lambda(\mathcal{S}^{n,k})$ of eigenvalue vectors of all such matrices is
star-shaped with respect to the nonnegative orthant $\mathbb{R}^n_{\geq 0}$ and
not convex already when $(n,k)=(4,2)$.
