It is stated that any polytope is an affine projection of a simplex. I do not quite understand this:
on the plane, a simplex has exactly 3 vertices, but let's consider a polytope $P=\{(x,y)\mid 0\leq x\leq 1, 0\leq y\leq 1\}$ which has 4 vertices, which simplex an affine projetion of which is the polytope $P$?
I assume I am wrong, but where is the problem?
You need to be allowed to use simplices whose dimension is higher than your target polytope. In the example you give, you would need a three-dimensional simplex (which has 4 vertices).