This paper deals with the study of a class of variational problems under convexity constraints. These problems are related to Newton’s minimal resistance problem, as well as to recent problems arising in the theory of incentives in Mathematical Economics. The author first proves a compactness result which enables him to establish the existence of solutions for nonconvex Lagrangians or energy functionals with linear growth. Then a first order necessary condition is established by a penalization method. In the last part of the paper there are given regularity results in dimension one and in the radial case. The author also makes a comparison between the convex envelope and the solution of a projection variational problem under a convexity constraint.