Dimension Formula

Find a formula for the dimension of the resultant polytope $\mathcal{A}$ in terms of $n$ and $m$.
Hint: What is the dimension of the lineality space of the normal fan?

## Solution:

The ambient space has dimension $m$, since this is the number of vertices in all the polytopes, each of which must be assigned a weight. There is a linear space of dimension 1 for each polytope which corresponds to translations of all the points. Additionally there is a linear space of dimension $n$ corresponding to linear transformations on all of the polytopes together in each of the $n$ directions. The resultant polytope therefore sits in a codimension $n+1+n$ subspace, so has dimension $m-2n-1$.