An inequality concerning edges of minor weight in convex 3-polytopes (Q2785651)

Using the theory of connected planar maps and continuing investigations of B. Grünbaum, E. Jucovič, O. V. Borodin and others, the authors give the final answer to a conjecture raised by Grünbaum (1973) and referring to edges of minor weight of convex polyhedra \(P\) in 3-space. Denoting by \(e_{ij}\) the number of edges of \(P\) joining the vertices of degree \(i\) with the vertices of degree \(j\), they show that the inequality NEWLINE\[NEWLINE\begin{multlined} 20e_{3,3} +25e_{3,4} +16e_{3,5} +10e_{3,6} +6 {2\over 3} e_{3,7} +5e_{3,8} +2 {1\over 2} e_{3,9}+ \\ +2 e_{3,10} +16 {2\over 3} 3_{4,4} +11e_{4,5} +5e_{4,6} +1 {2\over 3} e_{4,7} +5 {1\over 3} e_{5,5} +2 e_{5,6} \geq 120 \end{multlined}NEWLINE\]NEWLINE holds. Moreover, it is verified that here each coefficient is the best possible.