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

The basis number of a graph $G$ is defined by Schmeichel to be the least integer $h$ such that $G$ has an $h$-fold basis for its cycle space. MacLane showed that a graph is planar if and only if its basis number is $\le 2$. Schmeichel proved that the basis number of the complete graph $K_n$ is at most $3$. We generalize the result of Schmeichel by showing that the basis number of the $d$-th power of $K_n$ is at most $2d+1$.