The Minimal Dimensionality of Stable Faces Required to Guarantee Stability of a Matrix Polytope

We consider the problem of determining whether a polytope of n×n matrices is stable, by checking stability of low-dimensional faces of the polytope. We show that stability of all (2n-4)-dimensional faces guarantees stability of the entire set. Furthermore, we prove that, for any n and any k¿2n-4, there exists an unstable polytope of dimension k such that all its (2n-5)-dimensional subpolytopes are stable.