Let $C \subset \mathbb{R}^{n}$ be a closed convex cone. If one want to know whether the linear map $T:\mathbb{R}^{n} \mapsto\mathbb{R}^m$ sends the closed set $C$ to another closed one, $T(C)$, it's needed to prove that $\text{ker } T + C$ is closed.

My concern turns to know whether there exist good examples of cones other than polyhedral cones that are mapped to closed sets by linear maps. Hence, the questions is reduced to:

There exist a closed convex cone $C$, different from any polyhedral cone, such that, for every vector subspace $V$, $V+C$ is closed?

The article On the Closedness of the Linear Image of a Closed Convex Cone almost answers my question. If one can find an enlightening example of cone $C$ satisfying the condition (SUM-WE), my example is constructed.