Are the following sets convex cones:
1.$\{x\in\Bbb R^n:\langle a,x\rangle\leq 0, a\neq 0 \}$
2.$\{x\in\Bbb R^n:\langle a,x\rangle\lt 0, a\neq 0 \}$ ?
We say $C\subset \Bbb R^n$ is a cone if $\forall x \in C$ and $\lambda>0$, $\lambda x \in C$. A convex cone is a cone that is a convex set.
I drew the sets in $\Bbb R^2$ and from that we can see that both sets are convex cones. Is there a more formal proof?