subsection is to prove the following theorem.

\[thm:existence-convex-poly\] If the switched linear system (\[eq:switched.linear.system\]) is asymptotically stable under arbitrary switching, then there exists a convex positive definite homogeneous polynomial $p$ that satisfies $p(A_ix)<p(x)$ for all $x\neq 0$ and all $i\in\{1,\ldots,m\}.$

Our proof is inspired by [@mason-boscain-chitour06], which proves the existence of a convex polynomial Lyapunov function for continuous time switched systems, but we are not aware of an equivalent statement in discrete time. We also need the following classical result, which to the best of our knowledge first appears in [@RoSt60].

\[thm-rotastrang\] Consider a set of matrices ${{\cal{M}}}\subset {{\mathbb R}}^{n\times n}$ with JSR $\rho$. For all $\epsilon>0$, there exists a vector norm $|\cdot|_\epsilon$ in ${{\mathbb R}}^n$ such that for any $A$ in ${{\cal{M}}},$ $$|x|_\epsilon \leq 1 \quad \Rightarrow \quad |Ax|_\epsilon \leq \rho + \epsilon.$$

(of Theorem \[thm:existence-convex-poly\].) Let ${{\cal{M}}}\mathrel{\mathop:}=\{A_1,\ldots,A_m\}$ and denote the JSR of ${{\cal{M}}}$ by $\rho$. By assumption we have $\rho<1$ and by Theorem \[thm-rotastrang\], there exists a norm, which from here on we simply denote by $|\cdot|$, such that $\forall i\in\{1,\ldots,m\},$ $$|x| \leq 1  \Rightarrow \quad |A_ix| \leq \rho + \frac12 (1-\rho).$$ We denote the unit ball of this norm by $B$ and use the notation $${{\cal{M}}}B\mathrel{\mathop:}=\{A_ix|~A_i\in{{\cal{M}}}\mbox{ and 