Convex Bodies of Optimal Shape

Given a continuous function f: S^(n-1) --> R, we consider the minimization of the functional Integral over partial A of f (n_(A)) dH^(n-1) with respect to the subset A of R^(n), included in a class of convex bodies defined by surface or shape conditions. This corresponds to non-parametric formulations of older problems, including Newton's problem of the body of minimal resistance, following an approach due to G. Buttazzo and P. Guasoni [J. Convex Analysis 4 (1997) 343--351]. We establish existence and uniqueness results and some characterizations of the minimizers