Let $K_1$ and $K_2$ be two convex cones, including the origin, in a real vector space. Show that $K_1 + K_2 = \text{conv}(K_1 \cup K_2)$.
It is straight forward to show that $K_1 + K_2$ is a convex cone. Tow show the statement we need to proof $K_1 + K_2 \subseteq \text{conv}(K_1 \cup K_2)$ and $\text{conv}(K_1 \cup K_2) \subseteq K_1 + K_2$.
To show $K_1 + K_2 \subseteq \text{conv}(K_1 \cup K_2)$ we can write
Let $x \in K_1+ K_2$ and $y \in K_1+ K_2$, therefore
$\lambda x+(1-\lambda)y \in K_1+ K_2$ because $K_1 + K_2$ is a convex cone.
To show $\text{conv}(K_1 \cup K_2) \subseteq K_1 + K_2$
Let $z \in \text{conv}(K_1 \cup K_2)$.
How can we proceed?