Convex bodies with extremal volumes having prescribed brightness in finitely many directions

We consider the class of convex bodies in the n-dimensional Euclidean space with prescribed projection (n - 1)-volumes along finitely many fixed directions. We prove that in such a class there exists a unique body (up to translation) with maximum n-volume. The maximizer is a centrally symmetric polytope and the normal vectors to its facets depend only on the assigned directions. Conditions for the existence of bodies with minimum n-volume in the class defined above are given. Each minimizer is a polytope, and an upper bound for the number of its facets is established