Finite Combinatorial Types

Show there are finitely many combinatorial types of resultant polytopes in each dimension.

## Solution:

Using the dimension formula, resultant polytopes with dimension $d$ satisfy $m = 2(n+1) + d - 1$. For each $i$, $|A_i| \geq 2$ and by exercise 2, if $|A_i| = 2$ we can eliminate $A_i$ and the dimension of the resultant polytope will be unchanged. After removing all such unnecessary polytopes, each remaining one must have at least 3 vertices, so $n+1 \leq d-1$. For a fixed $d$ the combinatorial types are fully determined by how the "extra" $d-1$ vertices are distributed among $A_1,\ldots,A_{n+1}$ with each one receiving at least one, and there are only a finite number of ways to do this.