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?



When does every point in a polytope lie along a chord between its edges?
The question asks whether every point $v$ in the interior of a 3-polytope $P \ $ is on an interval between two edge-points. This is easy. Project the edges of $P$ onto a unit sphere centered at $v$. Call the resulting graph $G$ *blue*. Take the opposite $-G$ and call this graph *red*. Clearly red and blue graphs intersect, since otherwise one must lie in the face of another, which is impossible since $v$ is interior. Thus the line through the intersection point and $v$ is as desired. 


As for higher dimensions, this is clearly not possible already for dim-reasons. We are talking about 2-parametric family of intervals, which cannot possibly cover the interior of a $d$-polytope, for $d\ge 4$. 


