This is a reference request for a description of polytopes by affine relations of their vertices. Let me give a few examples of what I'm looking for:
Taking $n+1$ points $a_0,\dots,a_n$ in $\Bbb R^d$ that are affinely independent (i.e., $a_i-a_0$ for $i=1,\dots,n$ are linearly independent), the convex hull $$ \Delta^n = \operatorname{conv}\{a_0,\dots,a_n\} $$ is an $n$-simplex. We might say $n$-simplices are exactly those polytopes with $n$ vertices and no affine relations on the vertices.
Now consider a parallelogram $P=\operatorname{conv}\{a,b,c,d\}\subset\Bbb R^d$ where $a$ and $d$ are opposing vertices. This will satisfy the affine relation $$ a+d = b+c. $$ So we might say that parallelograms are exactly those polytopes with $4$ vertices that satisfy (only) the affine relation $a+d=b+c$.
It looks like this line of thought should yield a characterization of affine equivalence classes of polytopes by the affine relations of their vertices.
Another way to phrase this would be to take a polytope $P=\operatorname{conv}\{a_1,\dots,a_n\}\subset\Bbb R^d$ and associate to it the subspace $U_P\subseteq \Bbb R^n$ given by the equations $\sum_i c_i x_i = 0$ whenever $\sum_i c_i a_i = 0$ and $\sum_i c_i=0$. I expect that $U_P$ determines the affine type (and a vertex labelling) of $P$.
I would assume this has been studied. Can somebody give me some pointers on what I might be looking for?