I am currently working on a problem for which I believe the following result is crucial.
The result of this problem was discussed in this post.
Product of simplicial complexes?
However it is not clear to me why this construction works.
The definition of a simplicial complex that is used in the source of this problem is a collection of simplices in $\mathbb R^n$ which includes faces of its elements and such that its elements either intersect to give $\emptyset$ or another simplex in the collection.
Here it is obvious that the vertices in question are affinely independent, and that faces of elements of the collection are included in it, but I have not been able to show that the collection is stable under taking intersections.