Continuity of the combinatorial structure of a polytope with respect to face variables

Question

Suppose we are given a convex polytope in terms of the face variables.

That is, let $Y = (1,x_1,\dots,x_n)$ and suppose we have vectors $W_a$ in $\mathbb{R}^{n+1}$ such that the locus $W_a \cdot Y \ge 0$ defines a convex polytope in $\mathbb{R}^n$. The vectors $W_a$ are the face variables.

Is it possible to change slightly these $W_a$ in such a way that the locus $W_a \cdot Y \ge 0$ defines a polytope combinatorially equivalent to the original one? (Same facet poset)

In this sense, how "continuous" is the combinatorial structure of a polytope as a function of the face variables?

Answer 1

If you apply an affine, or more generally, a projective transformation to $\mathbb{R}^n$, then the image of the polytope will have the same combinatorics, while the "face variables" will change in a non-trivial way.

In some cases this is the only possibility: there are polytopes whose combinatorics determines them up to a projective transformation, see for example

Adiprasito, Karim A.; Ziegler, Günter M., Many projectively unique polytopes, Invent. Math. 199, No. 3, 581-652 (2015). ZBL1339.52011.