Given a *finite* set of *convex* $d$-dimensional polytopes $\mathcal P$, for some $d\ge 2$.

> **Question:** Is it true that there are only *finitely* many different *convex* $(d+1)$-dimensional polytopes whose facets are solely (uniformly scaled and rotated versions of) polytopes in $\mathcal P$?

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**Some clarifications**

A face of a polytope is the intersection of the polytope with a touching hyperplane, so subdividing facets does not count here.

In general, two $(d+1)$-polytopes shall be considered as different if they differ not just in scale and orientation.
By the usual [rigidity arguments](https://en.wikipedia.org/wiki/Cauchy%27s_theorem_(geometry)), given the shape of facets and their connections, the metric of the polytope is uniquely determined. Hence, if we can built different polytopes, they will be combinatorially different as well.

**Example**

There are only finitely many polyhedra that can be built from any finite set of *regular* polygons, but as far as I know, this result is by enumeration  (see, e.g. [Johnson solids](https://en.wikipedia.org/wiki/Johnson_solid)).