TITLE: Is projection of a convex polyhedron on a plane a convex polygon?
QUESTION [5 upvotes]: If we have a convex polyhedron with vertices $\mathbf{V}$ and project it on a plane $\mathbf{P}$, is this procedure equivalent to projecting points in $\mathbf{V}$ on the plane $\mathbf{P}$ and then computing the 2D convex hull of the projected vertices? In other words, is the projection of a convex polyhedron on a plane a convex polygon? If so, how we can prove it mathematically?

REPLY [2 votes]: If $P$ is any linear map and
$$
x = \sum_i \lambda_i x_i
$$
then
$$
Px = \sum_i \lambda_i Px_i
$$
which means: yes! Convex combinations are preserved under any linear map.