# Are the polyhedral cones the only examples of cones that remains closed when they are added to vector subspaces?

Let $$C \subset \mathbb{R}^{n}$$ be a closed convex cone. If one wants to know whether the linear map $$T:\mathbb{R}^{n} \to\mathbb{R}^m$$ sends the closed set $$C$$ to another closed one, $$T(C)$$, it is needed to prove that $$\text{ker } T + C$$ is closed.

My concern turns into to know whether there exist good examples of cones, other than polyhedral cones, that are mapped to closed sets by any linear map. Hence, the question 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) for every subspace, my example is constructed.

• Since the linked article is behind a paywall it might be useful to formulate explicitely the SUM-WE condition. May 25, 2022 at 7:26
• @GuillaumeAubrun, there is a question by a student who heard one of your talks, something about a cone in $\mathbb R^3$ with a dense set of planar cross sections. math.stackexchange.com/questions/4469943/… Is there a standard reference for this sort of thing? Jun 10, 2022 at 21:40
The radial cone of $$C$$ is defined via $$\mathcal R_C(x) := \bigcup_{\lambda > 0} \lambda ( C - x)$$ for all $$x \in C$$ and we can show $$\mathcal R_C(x) = C + \operatorname{span}(x),$$ since $$C$$ is a cone. By assumption $$\mathcal R_C(x)$$ is closed for all $$x \in C$$, since $$\operatorname{span}(x)$$ is a subspace. But now, Proposition 2 of Duality of linear conic problems by Shapiro and Nemirovski implies that $$C$$ is a polyhedral cone.