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On the convexity of sublevel sets of polynomial and homogeneous polynomial Lyapunov functions

Polynomial and homogeneous polynomial Lyapunov functions have recently received a lot of attention from the control community. However, no condition is still available to establish if their sublevel sets are convex, property that can be useful in several applications. This paper proposes some conditions to investigate this convexity property based on polynomial relaxations that can be handled through convex linear matrix inequalities optimizations