My question is actually a reference request, but being unfamiliar with the topic, pointing me towards the right reference would already be a big help. It can be broken down into two parts.
1) Let $\Delta_n=\{x \in \mathbb R^{n+1}|x_0+x_1+\dots+x_n=1,\,x_i\geq 0\}$ be the standard $n$-simplex. Let $L:\mathbb R^{n+1}\to \mathbb R^m$, be a general linear mapping. Is it true that $L(\Delta_n)$ is a convex polytope, i.e. the convex hull of a finite number of vertices?
2) If yes, how to compute its vertices? I guess that if $n+1=m$ and $L$ is invertible, I can compute them as the linear image under $L$ of each of the vertices of the simplex. But in the general case where $m\leq n$, is there a way to compute them from knowledge of the vertices of $\Delta_n$? If not, how to find the H-representation of $L(\Delta_n)$?
1) Yes. I don't know offhand of a reference for this, but it's a basic fact that the image under any linear map of any convex polytope is another convex polytope. One way to characterize the points of a convex polytope is that they are linear combinations of the vertices, with non-negative coefficients which sum to 1; then its image under the linear map is a linear combination of the images of the vertices, with the same coefficients.
2) The vertices of the image are a subset of the images of the original vertices.