that $V$ is sos; see [@Pablo_Jadbabaie_JSR_journal Thm. 2.2]. (This statement fails to hold for switched nonlinear systems.)

In the next section, we study the converse question of existence of a Lyapunov function satisfying the semidefinite conditions in (\[eq:sos-convex.Lyap.requirements\]).

Sos-convex Lyapunov functions and switched linear systems {#section:linear}
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As remarked in the introduction, it is known that asymptotic stability of a switched linear system under arbitrary switching implies existence of a common convex Lyapunov function, as well as existence of a common polynomial Lyapunov function. In this section, we show that this stability property in fact implies existence of a common Lyapunov function that is both convex and polynomial (cf. Subsection \[subsec:existence.convex\]). Moreover, we strengthen this result to show existence of a common sos-convex Lyapunov function (cf. Subsection \[subsec:sos-convex\]).

Before we prove these results, we state a related proposition which shows that in the particular case of switched linear systems, any common Lyapunov function (e.g. a non-convex polynomial) can be turned into a common *convex* Lyapunov function, although not necessarily an efficiently computable one. We believe that this statement must be known, but since we could not pinpoint a reference, we include a proof here.

\[prop:Minkowski\] Consider the switched linear system 