A subset $C$ of a vector space is a cone if for any element $x$ of $C$ and for any non-negative scalar $\alpha$, $ \alpha x\in C$.
Let $C$ be a cone.
When the sum of any two elements of $C$ is also in $C$, then the cone is said to be convex
I say $C$ is "the opposite of a convex cone" if the sum of any two linearly independent vectors of $C$ is outside of $C$. For example, a right circular cone (as a surface) is just that. Is there a standard name for this?