On Nov 11, 2017, at 11:06 AM, James Propp <jamespropp@gmail.com> wrote:
Veit's proof collapses the 24 edge-meets-edge terms into a single term whose value is the constant 2! What's more, his argument can be used to show that for any centrally symmetric 2n-gon, the expected number of vertices in the intersection of the 2n-gon with a translate of itself, conditioned on the event that the intersection is nonempty, is exactly n+2. The part of his argument that I still don't understand is this: how do we know that (generically) the boundaries of the two 2n-gons intersect exactly twice?
Jim Propp
Here’s the proof, generalized to polytopes: P is a finite convex polytope in N dimensions. We want to know the topology of the intersection of the boundary of P and the boundary of P+v, where the translation vector v is not in any of the subspaces defined by the faces of P. 1. Let V be the subspace orthogonal to v and y the coordinate of R^N orthogonal to V. Orient y so that the y-coordinate of v, y(v), is positive. 2. Let X be the projection of P into V. 3. Consider the line L(x) parallel to v at an interior point x of X. Because of the face restriction on v, the intersection of L(x) and P is a pair of points, (x,y1(x)) and (x, y2(x)), where y2(x)>y1(x). This defines two continuous functions on X: the convex function y1(x) and the concave function y2(x). 4. The intersection of P and P+v is the set (x,y(x)), where x is in X and y1(x)+y(v)<=y<=y2(x). Assume henceforth the intersection is non-empty. The boundary of the intersection is the union of the sets C1: (x,y1(x)+y(x)) and C2: (x,y2(x)) where x is in X and satisfies y2(x)-y1(x)-y(v)>=0. C1 is a subset of the boundary of P+v and C2 is a subset of the boundary of P. 5. Because y2 is concave and y1 is convex, the function y2(x)-y1(x)-y(v) is concave on X. The subset of X, where y2(x)-y1(x)-y(v)>=0, therefore defines a closed convex set, X(v). 6. From the boundary of X(v), where y2(x)-y1(x)-y(v)=0, we have a continuous 1-1 map to the intersection of C1 and C2. 7. Since X(v) is a closed convex set its boundary is a topological sphere in N-1 dimensions. By 6. this is also the topology of the intersection of C1 and C2 which by 4. is the intersection of the boundaries of P and P+v. 8. N=2: The intersection of the boundaries of convex polygons P and P+v is a pair of points (a topological 1-sphere).