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This paper deals with switched linearsystems with persistent disturbances and under arbi-trary switching. For these systems, a systematic com-ponentwise ultimate bound computation method hasbeen previously developed. This method does not em-ploy a Lyapunov function, yet yields a mixture of ellip-soidal and polyhedral sets, which are the type of levelsets obtained via complex polytopic Lyapunov func-tions. In this context, our contribution is as follows:(a) we show that if the aforementioned component-wise method can be applied, then a complex polytopicLyapunov function exists based on which the same ul-timate bound is obtained; (b) we provide a novel al-gebraic condition for the existence of a complex poly-topic Lyapunov function of minimum complexity; (c)we give an example for which a polytopic Lyapunovfunction exists but the componentwise method cannotbe applied. These results serve to establish the preciseconnection between the two approaches.