Let $C$ and $D$ be closed convex cones in $R^n$. I am trying to show that $C\cap D$ is a closed cone.
I started with
Take any point $x_1 \in C$ and $x_2 \in D$ with $\theta_1,\theta_2\geq0$ and $\theta_1+\theta_2=1$.
After that I do not think I am doing it right. I think I am suppose to let $x=\theta_1x_1+\theta_2x_2$ and show that $x\in C\cap D$? Am I on the right track?
Best Answer
Start with $x_1,x_2 \in C \cap D.$ Now, this means that $x_1,x_2 \in C$. Because $C$ is a convex cone, we know that for any $\theta_1,\theta_2>0$ that $$\theta_1x_1+\theta_2x_2 \in C.$$ Similarly, $$\theta_1x_1+\theta_2x_2 \in D.$$ Hence, $$\theta_1x_1+\theta_2x_2 \in C\cap D$$ hence $C \cap D$ is convex. Now, just show it must also be a cone. Simply show that $\alpha > 0, x_1 \in C \cap D \Rightarrow\alpha x_1 \in C \cap D.$ This is straightforward.