Angles as probabilities

arXiv:0809.3459v1 [math.MG] 19 Sep 2008 Angles as probabilities David V. Feldman Daniel A. Klain Almost everyone knows that the inner angles of a triangle sum to 180o. But if you ask the typical mathematician how to sum the solid inner angles over the vertices of a tetrahedron, you are likely to receive a blank stare or a mystified shrug. In some cases you may be directed to the Gram-Euler relations for higher dimensional polytopes [3, 4, 6, 7], a 19th century result unjustly consigned to relative obscurity. But the answer is really much simpler than that, and here it is: The sum of the solid inner vertex angles of a tetrahedron T, divided by 2π, gives the probability that the orthogonal projection of T onto a random 2-plane is a triangle. How simple is that? We will prove a more general theorem (Theorem 1) for simplices in Rn, but first consider the analogous assertion in R2. The sum in radians of the angles of a triangle (2-simplex) T, when divided by the length π of the unit semicircle, gives the probability that the orthogonal projection of T onto a random line is a convex segment (1-simplex). Since this is always the case, the probability is equal to 1, and the inner angle sum for every triangle is the same. By contrast, a higher dimensional n-simplex may project one dimension down either to an (n −1)-simplex or to a lower dimensional convex polytope having n + 1 vertices. The inner angle sum gives a measure of how often each of these possibilities occurs. Let us make the notion of “inner angle” more precise. Denote by Sn−1 the unit sphere in Rn centered at the origin. Recall that Sn−1 has (n −1)-dimensional volume (i.e. surface area) nωn, where ωn = πn/2 Γ(1+n/2) is the Euclidean volume of the unit ball in Rn. Suppose that P is a convex polytope in Rn, and let v be any point of P. The solid inner angle aP(v) of P at v is given by aP(v) = {u ∈Sn−1 | v + ǫu ∈P for some ǫ > 0} Let αP(v) denote the measure of the solid angle aP(v) ⊆Sn−1, given by the usual surface area measure on subsets of the sphere. For the moment we are primarily concerned with values of αP(v) when v is a vertex of P. If u is a unit vector, then denote by Pu the orthogonal projection of P onto the subspace u⊥in Rn. Let v be a vertex of P. The projection vu lies in the relative interior of Pu if and only if there exists ǫ ∈(−1, 1) such that v + ǫu lies in the interior of P. This holds iffu ∈±aP(v). If u is a random unit vector in Sn−1, then (1) Probability[vu ∈relative interior(Pu)] = 2αP(v) nωn , This gives the probability that vu is no longer a vertex of Pu. 1 2 For simplices we now obtain the following theorem. Theorem 1 (Simplicial Angle Sums). Let ∆be an n-simplex in Rn, and let u be a random unit vector. Denote by p∆the probability that the orthogonal projection ∆u is an (n −1)-simplex. Then (2) p∆= 2 nωn X v α∆(v) where the sum is taken over all vertices of the simplex ∆. Proof. Since ∆is an n-simplex, ∆has n + 1 vertices, and a projection ∆u has either n or n + 1 vertices. (Since ∆u spans an affine space of dimension n −1, it cannot have fewer than n vertices.) In other words, either exactly 1 vertex of ∆ falls to the relative interior of ∆u, so that ∆u is an (n −1)-simplex, or none of them do. By the law of alternatives, the probability p∆is now given by the sum of the probabilities (1), taken over all vertices of the simplex ∆. □ The probability (2) is always equal to 1 for 2-dimensional simplices (triangles). For 3-simplices (tetrahedra) the probability may take any value 0 < p∆< 1. To obtain a value closer to one, consider the convex hull of an equilateral triangle in R3 with a point outside the triangle, but very close to its center. To obtain p∆close to zero, consider the convex hull of two skew line segments in R3 whose centers are very close together (forming a tetrahedron that is almost a parallelogram). Similarly, for n ≥3 the solid vertex angle sum of an n-simplex varies within a range 0 < X v αT(v) < nωn 2 . Equality at either end is obtained only if one allows for the degenerate limiting cases. These bounds were obtained earlier by Gaddum [1, 2], using a more com- plicated (and non-probabilistic) approach. Similar considerations apply to the solid angles at arbitrary faces of convex polytopes. If F is a face of a convex polytope P, denote the centroid of F by ˆF, and define the solid inner angle measure αP(F) = αP( ˆF), using the definition above for the solid angle at a point in P. In analogy to (1), we have (3) Probability[ ˆFu ∈relative interior(Pu)] = 2αP(F) nωn , Omitting cases of measures zero, this gives the probability that a proper face F is no longer a face of Pu. (Note that dim Fu = dim F for all directions u except a set of measure zero.) Taking complements, we have (4) Probability[Fu is a proper face of Pu] = 1 −2αP(F) nωn . For 0 ≤k ≤n −1, denote by fk(P) the number of k-dimensional faces of a polytope P. The sum of the probabilities (4) gives the expected number of k-faces 3 of the projection of P onto a

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