On the structure of the set of extremal norms of a linear inclusion

A systematic study of the set of extremal norms of an irreducible linear inclusion is undertaken. We recall basic methods for the construction of extremal norms, and consider the action of basic operations from convex analysis on these norms. It is shown that the set of extremal norms of an irreducible linear inclusion is a convex cone with a compact basis in an appropriate Banach space. Furthermore, the compact basis may be chosen to depend upper semi-continuously on the data. We explain that this is the reason for the local Lipschitz continuity of the joint spectral radius as a function of the data.