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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?

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Hint: A set $\mathbf{P}$ is convex iff for any points $x,y\in \mathbf{P}$ the segment $[x,y]$ joining them belongs to $\mathbf{P}$.

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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.

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