Stability of a polytope of matrices: counterexamples

While there have been significant breakthroughs for the stability of a polytope of polynomials since V.L. Kharitonov's (1978) seminal result on interval polynomials, for a polytope of matrices, the stability problem is considered far from completely resolved. Counterexamples are provided for three conjectures that are directly motivated by the results in the polynomial case. These counterexamples illustrate the fundamental differences between polynomial-stability and matrix-stability problems and indicate that some obvious lines of attack on the matrix polytope stability problem will fail.>