Proof that the projection of a convex body is a convex?

Question

In general i have found that the projection of a convex body is also convex, but i can't find the proof of it. Speciffically the simplest of case: prove that the 2d projection of a convex 3d body is also convex.

Answer 1

If $C$ is convex and $T$ is a linear map, then the image $T[C]$ is convex, since $$ \lambda Tx + (1-\lambda) Ty =T(\lambda x + (1-\lambda)y)\in T[C] $$ But projections are linear.