Information needed to distinguish combinatorially isomorphic polytopes (up to affine equivalence)

Question

I originally posted this question on Stack Exchange, thinking it perhaps does not qualify as "research-level" but it received no answers... hopefully someone here can help.

The title pretty much sums up the question: what extra information do we need (or what is an example of sufficient information) on top of the face lattice in order to characterize a convex polytope, up to affine transformations? For example, would it be sufficient to provide a list of normal vectors associated with each facet for some embedding in $\mathbb{R}^d$?

Answer 1

There is characterization of a (convex) polytope in terms of facet normals and facet areas. It's called Minkowski's theorem (which isn't the most unique name, I guess) and it goes as follows:

Let $u_1, \ldots, u_n$ be unit vectors which span $\mathbb{R}^d$, and let $a_1, \ldots, a_n > 0$ satisfy $a_1 u_1 + \ldots + a_n u_n = 0$. Then there exists up to translation a unique convex polytope with $n$ facets $F_1, \ldots, F_n$ such that $\mathrm{area}(F_i) = a_i$ and the normal to $F_i$ is $u_i$.

And if you're given a convex polytopes, the unit facet normals together with the facet areas satisfy the above equality as well.

A very nice writup of all of this and much more, can be found in Igor Pak's notes here. The formulation above is stolen from there too.

Answer 2

The relevant objects are slack matrices. Let $P$ be a convex polytope inside an affine space, with vertex set $V$ and facet set $F$. For any facet $f$, choose an affine form $\varphi_f$ which is $0$ on $f$ and $>0$ on vertices outside $f$. The matrix $S=(\varphi_f(v))$ indexed by $F \times V$ is called a slack matrix of $P$. Then

1. up to combinatorial equivalence, the zero pattern of $S$ determines $P$
2. up to projective equivalence, $S$ determines $P$ modulo multiplying each row and each column by a positive number
3. up to affine equivalence, $S$ determines $P$ modulo multiplying each row by a positive number

This is explained for example in Gouveia, João; Macchia, Antonio; Thomas, Rekha R.; Wiebe, Amy, The slack realization space of a polytope, SIAM J. Discrete Math. 33, No. 3, 1637-1653 (2019). ZBL1423.52032.

Answer 3

Here is a very simple example to show the kind of thing that can happen.

Suppose that $x>1$ and $-1<y<1$, and let $Q(x,y)$ denote the pentagon in $\mathbb{R}P^2$ with vertices as follows: \begin{align*} u_0 &= (1,1,1) \\ u_1 &= (-1,1,1) \\ u_2 &= (-1,-1,1) \\ u_3 &= (1,-1,1) \\ u_4 &= (x,y,1). \end{align*} One can check that every pentagon in $\mathbb{R}P^2$ is projectively equivalent to $Q(x,y)$ for a unique pair $(x,y)$. (There is a proof at http://neil-strickland.staff.shef.ac.uk/misc/pentagon.pdf, together with a brief indication of why this is relevant to the structure of the Stasheff operad, which explains my interest in the question.) Presumably, the answer for affine equivalence is not too different.