Antinorms on cones: theory and applications to dynamical systems

The concept of antinorm (a concave nonnegative homogeneous functional on a cone) was introduced in early 90s and found many applications in the theory of stability of linear dynamical systems. We begin with a theoretical overview. The main facts of the convex analysis, in particular, the Fenchel - Moreau theorem, stays true for antinorms, however, there are significant differences. In particular, there exist infinitely many self-dual antinorms and even self-dual polyhedral antinorms, which is not the case for norms. Then we demonstrate applications of antinorms to the problem of stabilization of a linear switching system and to the computation of a multiplicative Lyapunov exponent of random matrix products. Applications to the lower spectral radius of nonnegative matrices and to convex trigonometry are also addressed.