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 functi on f is not convex, the set Omega is a domain in R-2 and the minimum is sou ght over all convex functions on Omega with values in a given bounded inter val. We prove that a minimum u is almost everywhere 'on the boundary of con vexity', 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 zer o. 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 Om ega 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.