A note on interlacing polynomials (Q1813502)

Let \(A\) be an \(n \times n\) Hermitian matrix with eigenvalues \(\lambda_ 1 \geq \cdots \geq \lambda_ n\). For \(1 \leq k \leq n\), denote by \(\text{Int} (A;k)\) the set of monic degree-\(k\) polynomials whose roots interlace the eigenvalues of \(A\). For \(k = 1\), \(n - 1\), \(n\), it is known that \(\text{Int} (A;k)\) is convex, and in this paper it is shown that, for the remaining cases, \(\text{Int} (A;k)\) is convex if and only if \(\lambda_ 2 = \cdots = \lambda_{n-1}\).