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Nontrivial examples of cones $C$ that remains closed when they are added to vector subspaces

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 $\text{kern } T + C$ 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 $C$, other than polyhedral cone, such that, for every vector subspace $V$, $V+C$ is closed?