Convex Bodies of Optimal Shape

Given a continuous function $f\colon S^{n-1}\to\mathbb{R}$, we consider the minimization of the functional $\int_{\partial A} f(\nu_A)\,d\mathcal{H}^{n-1}$ with respect to the subset A of $\mathbb{R}^n$, 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