TITLE: Is every polytope combinatorially equivalent to the intersection of a simplex and a linear subspace?
QUESTION [7 upvotes]: I wonder whether such a result is known, and if so, whether the proof is trivial. 
By polytope I mean the convex hull of finitely many points in $\Bbb R^n$. Assume the simplex to be symmetric and centered at the origin, so that the subspace goes through its center. The subspace can have any dimension $k\in\{0,...,n\}$.

Question: Is every polytope combinatorially equivalent to the intersection of a simplex and a linear subspace through its center?

If this is true, are there estimations for the dimension of the simplex needed?

REPLY [10 votes]: The answer is yes.
The fact that any polytope is affinely equivalent to a section of a simplex is well-known (see the answer by Tobias Fritz).
Now in any simplex with vertices $(v_i)$ we may consider projective transformations via reweighting barycentric coordinates: given positive numbers $(a_i)$, such a transformation is given by
$$ \sum \lambda_i v_i \mapsto \frac{\sum a_i \lambda_i v_i}{\sum a_i \lambda_i}.$$
Projective transformations act transitively on the interior of the simplex, so it follows that any polytope is combinatorially equivalent to a section of some regular simplex through its center.
What I don't know is whether every polytope is affinely equivalent to a section of some regular simplex through its center.