  The joint spectral radius of a bounded set of d times d real or complex
matrices is defined to be the maximum exponential rate of growth of products of
matrices drawn from that set. Under quite mild conditions such a set of
matrices admits an associated vector norm, called a Barabanov norm, which can
be used to characterise those sequences of matrices which achieve this maximum
rate of exponential growth. In this note we continue an earlier investigation
into the problem of determining when the Barabanov norm associated to such a
set of matrices is unique. We give a new sufficient condition for this
uniqueness, and provide some examples in which our condition applies. We also
give a theoretical application which shows that the property of having a unique
Barabanov norm can in some cases be highly sensitive to small perturbations of
the set of matrices.
