analysis of the performance (existence) of convex Lyapunov functions.

On the optimization side, the reader interested in knowing more about sos-convex polynomials, their role in convex algebraic geometry and polynomial optimization, and their applications outside of control is referred to the works by Ahmadi and Parrilo [@AAA_PP_not_sos_convex_journal], [@AAA_PP_table_sos-convexity], Helton and Nie [@Helton_Nie_SDP_repres_2], and Magnani et al. [@convex_fitting], or to Section 3.3.3 of the edited volume [@Convex_Alg_Geom_BOOK]. Finally, we note that a shorter version of the current paper with some preliminary results appears in [@aaa_raph_sosconvex_cdc] as a conference paper.

Organization and contributions of the paper
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The paper is organized as follows. In Section \[section:sos-convex\], we present the mathematical and algorithmic machinery for working with sos-convex Lyapunov functions and explain its connection to semidefinite programming. In Section \[section:linear\], we study switched linear systems. We show that given any homogeneous Lyapunov function, the Minkowski norm defined by the convex hull of its sublevel set is also a valid (convex) Lyapunov function (Proposition \[prop:Minkowski\]). We then show that any stable switched linear system admits a convex polynomial Lyapunov function (Theorem \[thm:existence-convex-poly\]). Furthermore, we give algebraic arguments to strengthen this result and prove existence of an sos-convex Lyapunov function (Theorem \[thm:existence-sos-convex-poly\]). While existence of 