Gram's Equation - A Probabilistic Proof

If v are the angles at the vertices v, v 2 V , of a convex n-gon in the plane, then P v2V v = (n?2) { known as Euclid's angle-sum equation. We normalize the full angle to 1, i.e. we set v = v =(2). Then the equation reads as X v2V v = n=2 ? 1 (1) This identity has a generalization to higher dimensions, called Gram's equation. We want to derive this identity by a probabilistic argument. Let us immediately start with the proof, the identity will emerge quite naturally. Consider a convex polytope P in 3-space, with vertex set V , edge set E and face set F. For a vertex v, let v denote the fraction of an innnitesimally small sphere centered at v that is contained in P. Similarily, for an edge e, let e denote the fraction of an innnitesimally small sphere centered at a point in the relative interior of e that is contained in P. Let us now investigate a random parallel orthogonal projection of P, i.e. we choose a random point uniformly distributed on S 2 , and we project P in the direction speciied by this point to a plane orthogonal to the direction. The projection is a convex polygon. What is the expected number of vertices we get? The probability that a vertex v will not project to a vertex in the projected polygon is 2 v. Thus the expected number of vertices is X v2V (1 ? 2 v) : (2) Similarly, the expected number of edges in the projection is X e2E (1 ? 2 e) : (3) Now, since the number of vertices equals the number of edges, (2) equals (3) and we have the equation X v2V v ? X e2E e = (jV j ? jEj)=2 = ?jFj=2 + 1 : (4) The formula was known to de Gua (1783) for the case of a tetrahedron. 1 Hopf attributes the result for 3-polytopes to Brianchon (1837), while Gr unbaum refers to Gram 3] for a rst proof of the result. Gr unbaum reports that Gram's paper was forgotten, and that meanwhile Dehn 1] and Poincar e 7] contributed to the subject. Hopf gave another simple proof of the identity. He uses the angle-sum equation for spherical triangles (which we somehow replaced by the probabilistic argument). The formula generalizes to arbitrary dimensions (see e.g. …