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This paper presents a novel nonlinear sliding mode control methodology for systems with both matched and unmatched perturbations (including parametric uncertainties). Instead of traditional approaches where uncertainties and nonlinearities are coped with via linear nominal models and linear sliding surfaces, the proposed approach incorporates exact convex expressions to represent both the nonlinear surface and the system, thus allowing a significant chattering reduction. Moreover, thanks to the convex form of the nonlinear nominal model, when combined with the direct Lyapunov method, it leads to linear matrix inequalities, which are efficiently solved via convex optimization techniques. Illustrative examples are provided.