Inclusions**
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Difference inclusions, switched systems, nonlinear dynamics, convex Lyapunov functions, algebraic methods in optimization, semidefinite programming.

Introduction
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The most commonly used Lyapunov functions in control theory, namely the quadratic ones, are convex functions. This convexity property is not always purposefully sought after; it is simply an artifact of the nonnegativity requirement of Lyapunov functions, which for quadratic forms coincides with convexity. If one however seeks Lyapunov functions that are polynomial functions of degree larger than two (for instance, for improving some sort of performance metric), then convexity is no longer implied by the nonnegativity requirement of the Lyapunov function (consider, e.g., the polynomial $x_1^2x_2^2$). In this paper we ask the following question: what is there to gain (or to lose) by requiring that a polynomial Lyapunov function be convex? We also present a computational methodology, based on semidefinite programming, for automatically searching for convex polynomial Lyapunov functions.

Our study of this question is motivated by, and for the purposes of this paper exclusively focused on, the stability problem for difference inclusions, also known as discrete time switched systems. We are concerned with an uncertain and time-varying map $$x_{k+1}=\tilde{f_k}(x_k), \label{eq:switched.nonlinear.system}$$ where $$\tilde{f_k}(x_k)\in conv\{f_1(x_k),\ldots,f_m(x_k)\}.\label{eq:ftilda=conv}$$ Here, $f_1,\ldots,f_m: \mathbb{R}^n\rightarrow\mathbb{R}^n$ are $m$ different 