The minimal dimension of stable faces required to guarantee stability of a matrix polytope: D-stability

The author considers the problem of determining whether a polytope P of n*n matrices is D-stable, i.e. whether each point in P has all its eigenvalues in a given nonempty, open, convex, conjugate-symmetric subset D of the complex plane. The approach used is to check D-stability of certain faces of P. In particular, for each D and n the author determines the smallest integer m such that D-stability of every m-dimensional face guarantees D-stability of P.<>