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 $k\in\{0,...,n\}$.



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> **Question:** Is every polytope combinatorially equivalent to the intersection of a simplex and a linear subspace through its center?
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If this is true, are there estimations for the dimension of the simplex needed?



Is every polytope combinatorially equivalent to the intersection of a simplex and a linear subspace?
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.


