whether $\rho<1$ is notoriously difficult. No finite time procedure for this purpose is known to date, and the related problems of testing whether $\rho \leq 1$ or whether the trajectories of (\[eq:switched.linear.system\]) are bounded under arbitrary switching are known to be undecidable [@BlTi3]. On the positive side however, a large number of sufficient conditions for stability of such systems are known. Most of these conditions are based on the numerical construction of special classes of Lyapunov functions, a subset of which enjoy theoretical guarantees in terms of their quality of approximation of the joint spectral radius [@GP11; @Ando98; @Pablo_Jadbabaie_JSR_journal; @protasov-jungers-blondel09; @jungersguglielmicicone12].

It is well known that if the switched linear system (\[eq:switched.linear.system\]) is stable[^2], then it admits a common *convex* Lyapunov function, in fact a norm [@jungers_lncis]. It is also known that stable switched linear systems admit a common *polynomial* Lyapunov function [@Pablo_Jadbabaie_JSR_journal]. It is therefore natural to ask whether existence of a common *convex polynomial* Lyapunov function is also necessary for stability. One would in addition want to know how the degree of such convex polynomial Lyapunov function compares with the degree of a non-convex polynomial Lyapunov function. We address both of these questions in this paper.

It is 