Let A A be an n × n n \times n complex matrix. For 1 ≤ k ≤ n 1 \leq k \leq n we study the inclusion relation for the following polynomial sets related to the matrix A A . (a) The classical numerical range of the k k th compound of the matrix λ I − A \lambda I - A . (b) The k k th decomposable numerical range of the matrix λ I − A \lambda I - A . (c) The convex hull of the set of all monic polynomials of degree k k that divide the characteristic polynomial of A A . Moreover, we give an example showing that the set described in (a) is not convex in general. This settles a question raised by C. Johnson.