Is the intersection of a simplex a simplex?

Question

Let S be an n-simplex. So, intuitively, S's intersection with a subspace of R^n is either empty or another simplex. Can I say it for sure (ie, is the above statement provable)?

Answer 1

It is clearly not true that the intersection with R^(n-k) is an (n-k)-simplex.

Take n > 2, the subspace R^(n-1), and let the intersection consist of a single point - a vertex. The intersection is not an (n-1)-simplex, though it is a 0-simplex.

It is also not true that the intersection is always a simplex of any degree. It is possible to cut a tetrahedron with a plane and get a quadrilateral instead of a triangle. (Consider the plane with is parallel both to a given edge and the opposite edge)