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In this paper we discuss the notion of universality for classes of candidatecommon Lyapunov functions of linear switched systems. On the one hand, we provethat a family of absolutely homogeneous functions is universal as soon as itapproximates arbitrarily well every convex absolutely homogeneous function forthe $C^0$ topology of the unit sphere. On the other hand, we prove severalobstructions for a class to be universal, showing, in particular, that familiesof piecewise-polynomial continuous functions whose construction involves atmost $l$ polynomials of degree at most $m$ (for given positive integers $l,m$)cannot be universal.

On universal classes of Lyapunov functions for linear switched systems