TITLE: When does every point in a polytope lie along a chord between its edges?
QUESTION [10 upvotes]: Consider the 3-simplex, or tetrahedron, in 3-space.  Regardless of the positions of the vertices, every point in the simplex lies on a chord between two non-adjacent edges of the simplex.  Or, equivalently, every interior point lies along a straight line segment which intersects two non-adjacent edges.
When is this property true of other convex (or non-convex) polyhedra?  How does this property extend to the general $N$-simplex?

REPLY [3 votes]: I think that the proposed solution is slightly incomplete since the original question asks for nonadjacent edges of a 3-polytope. It can be easily fixed by studying the intersection of the graphs G and -G.  
Regarding possible n-dimensional versions of the problem, the following result can be proved. Let P be an n-dimensional polytope and k and m positive integers such that k + m = n + 1. For any point x in P there are two faces, F and G, of P such that dim F \le k - 1, 
dim G \le m - 1, and x is in conv(F U G).