Problem:
Define $A+B=\{a+b : a\in A, b\in B\}$. Suppose that the sum of overlapping convex cones $A_1, A_2, A_N\subseteq \mathbb{R}^{d}$ is such that $\mathbf{0}\in A_1\cap A_2\cap \ldots\cap A_N$. $$S = A_1+A_2+\ldots+A_N=A_1\cup A_2\cup \ldots\cup A_N,$$ where $S$ is a subspace. Under what conditions on the sets is this possible?
Context:
We know that the union of subspaces of $\mathbb{R}^{d}$ is a subspace iff one of them contains the other subspaces. I am curious about what happens if we relax the subspace constraint and replace it with convex cones. Since the convex cone assumption did not help me, I added one more condition (on the sum). Still, I am unable to solve it!