Lower bound theorems for general polytopes

For a d-dimensional polytope with v vertices, d+1≤v≤2d, we calculate precisely the minimum possible number of m-dimensional faces, when m=1 or m≥0.62d. This confirms a conjecture of Grünbaum, for these values of m. For v=2d+1, we solve the same problem when m=1 or d−2; the solution was already known for m=d−1. In all these cases, we give a characterisation of the minimising polytopes. We also show that there are many gaps in the possible number of m-faces: for example, there is no polytope with 80 edges in dimension 10, and a polytope with 407 edges can have dimension at most 23.