Given a closed convex cone $C \subset \mathbb{R}^n$ and a matrix $M \in \mathbb{R}^{m\times n}$, is the set $S = \{Mx\mid x \in C\}$ also a closed convex cone?
Firstly, $S$ must be a convex cone. But how about the closeness? I conjecture that $S$ must be closed if $C$ is a closed convex cone. However, I fail to come up with a rigorous proof (so maybe the claim is false). Note that for a more general $C$ (i.e., $C$ is not closed convex cone), the claim is not necessarily true.
Any hint or counterexample is really appreciated.
No, this is not true. You could take the rotated Lorentz cone $$C = \{(x,y,z) \in \mathbb{R}^3: y^2 \le x z; x \ge 0; z \ge 0\}$$ and project it onto the $y$-$z$-plane. The result is an open half-plane together with the origin.
Some conditions ensuring that the image is closed, can be found in this paper.