An example of non-convex minimization and an application to Newton's problem of the body of least resistance

We study the minima of the functional $int_Omega f(nabla u)$. The function $f$ is not convex, the set $Omega$ is a domain in $R^2$ and the minimum is sought over all convex functions on $Omega$ with values in a given bounded interval. We prove that a minimum $u$ is almost everywhere `on the boundary of convexity', in the sense that there exists no open set on which $u$ is strictly convex. In particular, wherever the Gaussian curvature is {finite, it is zero. An important application of this result is the problem of the body of least resistance as formulated by Newton (where $f(p) = 1/(1+abs{p^2)$ and $Omega$ is a ball), implying that the minimizer is not radially symmetric. This generalizes a result in~cite{bro.