Some necessary and sufficient conditions for low order interval polytopes to contain a Hurwitz polynomial

We establish some fundamental structure theorems which establish necessary and sufficient conditions for low order (less than eight) interval polytopes to contain a Hurwitz polynomial. For instance, an interval polytope of polynomials, generated by an interval polynomial of degree five, contains a Hurwitz polynomial if and only if an exposed two dimensional face of the polytope contains a Hurwitz polynomial. It turns out that these conditions are not uniform, in the sense, that these depend on the degree of the interval polynomial generating the interval polytope