Closedness of the sum of two cones

Question

Consider two closed convex cones $K_1$, $K_2$ in a topological vector space. It is known that, in general, the Minkowski sum $K_1 + K_2$ (which is the convex hull of the union of the cones) need not be closed.

Are there some conditions guaranteeing closedness of $K_1 + K_2$? It can be assumed that $K_1 \cap K_2 = \{0\}$ for simplicity.

Answer 1

A sufficient condition is that one of the cones is compact, since the following is true:

If $A,B$ are closed, and one of them even compact, then the Minkowski sum $A+B$ is closed. For a proof see for example this question.