Consider the 3-simplex, or tetrahedron, in 3-space. Regardless of the positions of the vertices, every point in the volume defined by the triangular faces of the polytope's skeleton graph can lie along a chord between two non-adjacent edges. Or, equivalently, every interior point can lie 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?