The authors investigate the discrete reliability for Crouzeix-Raviart finite element methods (FEMs). The paper outlines the optimality proof for nonconforming FEM for uniformly convex minimization problems and first provides the necessary preliminaries on regular triangulations into simples and their refinement in any space dimensions from [R. Stevenson, Math. Comput. 77, No. 261, 227–241 (2008; Zbl 1131.65095)]. The results are proved by means of a carefully designed transfer operator which is a discrete quasi interpolation for nonconforming finite element functions. Throughout the paper standard notation on Lebesgue and Sobolev spaces and their norms is employed. The authors conclude the paper with a sketch of the proof of the optimality of a convex minimization problem.