The following question came up in the context of an optimization problem.
Let $A \in \mathbb R^{m \times n}$ be a matrix and let $$K = \{x \in \mathbb R^n: Ax \le 0\}$$ be the polyhedral cone defined by the set of vectors in $\mathbb R^n$ that make a nonpositive angle with each of the rows of $A$.
Now consider the following polyhedral cone: $$K' = \{y \in \mathbb R^m: y^T A \ge 0\}.$$ $K'$ is the set of vectors in $\mathbb R^m$ whose angle with each of the columns of $A$ is nonnegative.
Question: Is there a name for $K'$? If not, what can we say about the (geometric) relation between the sets $K$ and $K'$?
I'm not aware of a term for the relationship between $K$ and $K'$. However, we can characterize $K'$ as the dual cone to the convex cone generated by the columns of $A$. That is, $K = C^*$, where $C$ is the convex cone defined by $C = \{Ax : x \in \Bbb R^n, x \geq 0\}$.
Similarly, $K$ is the polar cone to the convex cone generated by the rows of $A$.