An inequality concerning edges of minor weight in convex 3-polytopes

Let eij be the number of edges in a convex 3–polytope joining the vertices of degree i with the vertices of degree j. We prove that for every convex 3-polytope there is 20e3,3 +25e3,4 +16e3,5 +10e3,6 + 6 2 3 e3,7+5e3,8+2 1 2 e3,9+2e3,10+16 2 3 e4,4+11e4,5+5e4,6+1 2 3 e4,7+5 1 3 e5,5+ 2e5,6 ≥ 120; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.