I have a convex polyhedron (with integral nodes). I only calculate in euclidian spaces. Let N be the set of nodes, c the center (arithmetic mean) of the polyhedron.
I now want to determine if a line between two nodes is an edge of the polyhedron or is on a face of the polyhedron or not. In other words: i want to determine if there is a point on the line which isn't on the boundary of the polyhedron.
Let $ n,m \in N $ and $p^*=t^*\cdot n+(1-t^*)\cdot m$ the point with minimal distance to the center (i.e. $t^* = argmin_{t}(|p(t) - c|)$ ). I construct a plane through $p^*$ with normal vector $v=(c-p^*)$.
My intuition tells me that the line between $n$ and $m$ is an edge or a "diagonal" of a face of the polyhedron if and only if the plane does not split the polyhedron. In other words: if there exists no node $x \in N$ which satisfies $v\cdot (x-p^*)<0$, then the line is an edge of on a face.
Am I right about this? My mind isn't clear enough to see it right now. It's obvious in 2D, but my imaginations fails for higher dimensions (as I am dealing with at least six dimensions)