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

Let ${\displaystyle e_{ij}}$ 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 ${\displaystyle 20e_{3,3}+25e_{3,4}+16e_{3,5}+10e_{3,6}+6\left[\frac{2}{3}\right]e_{3,7}+5e_{3,8}+2\left[\frac{1}{2}\right]e_{3,9}+2e_{3,10}+16\left[\frac{2}{3}\right]e_{4,4}+11e_{4,5}+5e_{4,6}+1\left[\frac{2}{3}\right]e_{4,7}+5\left[\frac{1}{3}\right]e_{5,5}+2e_{5,6}\ge 120}$; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.