# When the convex hull of a closed convex cone and a ray is closed?

Let $$C$$ be a closed convex cone in a Hausdorff locally convex topological vector space $$E$$ and let $$y \in E$$. I wonder a condition under which the conical hull of $$C \cup \left\{ y \right\}$$, i.e., the set $$\left\{ x+\lambda y: x \in C, \lambda \geq 0 \right\}$$, is closed.

Any suggestions will be very much appreciated.

Let $$C^\ominus = \{ u\in E^* \,|\, \langle u,C\rangle \leq 0\}$$ and let $$H = \{u\in E^* \,|\, \langle u,-y\rangle \leq 0\}$$.
If the interior of $$C^\ominus$$ meets $$H$$, or if $$C^\ominus$$ meets the interior of $$H$$, then we have the sum rule $$\partial (\iota_{C^\ominus}+\iota_{H})= \partial \iota_{C^\ominus} + \partial \iota_{H}$$. In that case, evaluating at $$0$$, gives
$$(C^\ominus\cap H)^\ominus = C+H^\ominus = C+\mathbb{R}_+y\;\;\text{is closed.}$$
Another case arises when $$C$$ is a polyhedral cone. All the above is true in $$\mathbb{R}^n$$ and in Hilbert space, and likely in Banach space. I don't know about your very general setting.