Existence and uniqueness theorem for a 3-dimensional polytope of R^3with prescribed directions and perimeters of the facets

We give a necessary and su¢ cient condition for the existence and uniqueness up to translations of a 3-dimensional polytope P in R 3 having N facets with given unit outward normal vectors n 1 ; : : : ; n N and corresponding facet perimeters L 1 ; : : : ; L N. In this paper, a polytope of R 3 is the convex hull of …nitely many points in R 3. The classical Minkowski problem for polytopes in R 3 concerns the following question: Given a collection n 1 ; : : : ; n N of N pairwise distinct unit vectors in R 3 and F 1 ; : : : ; F N a collection of N positive real numbers, is there a polytope P in R 3 having the n i as its facet unit outward normals and the F i as the corresponding facet areas (1 i N), and, if so, is P unique up to translations? H. Minkowski proved that a polytope is uniquely determined, up to translations, by the directions and the areas of its facets (see [1, Theorem 9, p. 107]): Theorem 1 (H. Minkowski, 1897: [5] and [6, pp. 103-121]) A polytope in R 3 is uniquely determined, up to translations, by the directions and the areas of its facets. A well-known necessary condition for the existence of a polytope having facet unit outward normals n 1 ; : : : ; n N and corresponding facet areas F 1 ; : : : ; F N is that: 0 2010 MSC: 52B10; 52A2