Consider a convex polyhaedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhaedron antipodal if there exist two parallel planes passing through these points and such that the polyhaedron is contained between these planes. Let
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
be the number of antipodal pairs of vertices, and let
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
be the number of antipodal pairs of midpoint edges. Determine the difference
![A-B](/media/m/c/0/a/c0aa8e211ebb3bedf22c85ee74a22eca.png)
in terms of the numbers of vertices, edges, and faces.
%V0
Consider a convex polyhaedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhaedron antipodal if there exist two parallel planes passing through these points and such that the polyhaedron is contained between these planes. Let $A$ be the number of antipodal pairs of vertices, and let $B$ be the number of antipodal pairs of midpoint edges. Determine the difference $A-B$ in terms of the numbers of vertices, edges, and faces.