Lower bound theorems for general polytopes

Abstract For a d -dimensional polytope with v vertices, d + 1 ≤ v ≤ 2 d , we calculate precisely the minimum possible number of m -dimensional faces, when m = 1 or m ≥ 0 . 62 d . This confirms a conjecture of Grunbaum, for these values of m . For v = 2 d + 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.

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