in (\[eq:switched.linear.system\]). Suppose $V$ is a common homogeneous and continuous Lyapunov function for (\[eq:switched.linear.system\]); i.e. satisfies Let $$\mathcal{S}\mathrel{\mathop:}=\{x\in\mathbb{R}^n|~\ V(x)\leq 1\}.$$ Then, the Minkowski (a.k.a. gauge) norm defined by the set $\mathcal{S}$, i.e. the function $$W(x)\mathrel{\mathop:}=\inf \{t>0|\ \frac{x}{t}\in conv(\mathcal{S})\},$$ is a convex common Lyapunov function for (\[eq:switched.linear.system\]).

Since under the assumptions of the proposition the set $conv(\mathcal{S})$ is compact, origin symmetric, and has nonempty interior, the function $W$ is a norm (see e.g. [@BoydBook p. 119]) and hence convexity and positivity of $W$ are already established. It remains to show that for any $i\in\{1,\ldots,m\}$ and $x\neq 0$ we have $$\nonumber
\begin{array}{lll}
W(A_ix)&=& \inf \{t>0|\ \frac{A_ix}{t}\in conv(\mathcal{S})\}\\
\ &<& \inf \{t>0|\ \frac{x}{t}\in conv(\mathcal{S})\}\\
\ &=& W(x).
\end{array}$$ To see the inequality, first note that because $V$ is a common Lyapunov function, there must exist a constant $\gamma\in(0,1)$ such that if $z\in\mathcal{S}$, then $A_iz\in\gamma \mathcal{S}$. Now observe that if for some $t>0$ we have $\frac{x}{t}\in conv(\mathcal{S}),$ then by definition $\frac{x}{t}=\sum_j \lambda_j y_j$ for some $y_j\in\mathcal{S}$ and $\lambda_j\geq 0$ with $\sum_j \lambda_j =1.$ Hence, $$\frac{A_i x}{t}=\sum_j  \lambda_j A_iy_j=\sum_j  \lambda_j \gamma w_j,$$ for some $w_j\in\mathcal{S}$. But this means that $\frac{A_i x}{\gamma t}\in conv(\mathcal{S}).$

Existence of convex polynomial Lyapunov functions {#subsec:existence.convex}
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The goal of this 