  Consider continuous-time linear switched systems on R^n associated with
compact convex sets of matrices. When the system is irreducible and the largest
Lyapunov exponent is equal to zero, there always exists a Barabanov norm (i.e.
a norm which is non increasing along trajectories of the linear switched system
together with extremal trajectories starting at every point, that is
trajectories of the linear switched system with constant norm). This paper
deals with two sets of issues: (a) properties of Barabanov norms such as
uniqueness up to homogeneity and strict convexity; (b) asymptotic behaviour of
the extremal solutions of the linear switched system. Regarding Issue (a), we
provide partial answers and propose four open problems motivated by appropriate
examples. As for Issue (b), we establish, when n = 3, a Poincar\'e-Bendixson
theorem under a regularity assumption on the set of matrices defining the
system. Moreover, we revisit the noteworthy result of N.E. Barabanov [5]
dealing with the linear switched system on R^3 associated with a pair of
Hurwitz matrices {A, A + bcT }. We first point out a fatal gap in Barabanov's
argument in connection with geometric features associated with a Barabanov
norm. We then provide partial answers relative to the asymptotic behavior of
this linear switched system.
