On the Minkowski condition and divergence

Abstract The Minkowski condition for convex polytopes is equivalent to the divergence theorem of classical multivariable calculus. 1 The Minkowski condition for polytopes It is easy to see that a convex polygon in R2 is uniquely determined (up to translation) by the directions and lengths of its edges. This suggests the following, less easily answered, question in higher dimensions: given a collection of proposed facet normals and facet areas, is there a convex polytope in Rd whose facets fit the given data, and, if so, is the resulting polytope unique? This question, along with its answer, is known as the Minkowski problem. A convex polytope in Rd is defined to be the convex hull of a finite set of points in Rd. For a convex polytope P ⊆ Rd and a unit vector u ∈ Rd, denote by Pu the orthogonal projection of P onto the hyperplane u⊥, and denote by Pu the support set of P in the direction of u. Since P is a polytope, Pu will be a face of P. If dimP = n and Pu is a face of dimension n − 1 then Pu is called a facet of P, where u is the corresponding facet normal. The volume of a polytope P will be denoted by V (P). If P is a subset of a hyperplane in Rd, denote the (d − 1)-dimensional volume of P by v(P). More generally one could ask: Given a collection u1,..., uk of unit vectors and α1,..., αk> 0, unde