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 integral (Omega) f (delu). 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 + \p \ (2)) and Omega is a ball), implying that the minimizer is not radially symmetric. This generalizes a result in [1]. (C) 2001 Editions scientifiques et medicales Elsevier SAS.