This article presents a novel methodology for exact convex rewriting of nonlinear systems that generalizes the well-known sector nonlinearity approach; it allows improving the capabilities of those methodologies based on polytopes and the direct Lyapunov method, which are still an active research area of fuzzy control systems. Depending on the computational burden, a tighter fitting of interdependent nonlinearities allows avoiding useless or damaging vertices, thus remarkably improving the feasibility chances of analysis and design, while preserving conditions in the form of linear matrix inequalities. A variety of examples give account of the advantages of the proposal.