$[A^{-1}(\mathbb{R} ^m _{-})]^* = A^{T}(\mathbb{R} ^m _{+})$ polar cone

Question

I am trying to show that $[A^{-1}(\mathbb{R} ^m _{-})]^* = A^{T}(\mathbb{R} ^m _{+})$, where $A$ is a matrix, $A \in \mathbb{R} ^{m\times n}$, and $[A^{-1}(\mathbb{R} ^m _{-})]^*$ means polar cone of $[A^{-1}(\mathbb{R} ^m _{-})]$.

I don't even know how to start, would you please give me any hint or direction? thank you

Answer 1

Let $Q = \{ x \in \mathbb{R}^m | x \ge 0 \}$. Note that the dual (not polar) is $Q^* = Q$.

Note that $(A^{-1} (-Q))^\circ = (A^{-1} (Q))^*$.

In general, for a closed convex cone we have $C^{**} = C$, so we can show $(A^T(Q) )^* = A^{-1}(Q)$ instead

$(A^T(Q) )^* = \{ y | y^T A^T x \ge 0, \text{for all }x \in Q \} = \{ y | (Ay)^T x \ge 0, \text{for all }x \in Q \}$.

Note that $(Ay)^T x \ge 0 \text{ for all }x \in Q $ iff $Ay \in Q^* = Q$.

Hence $(A^T(Q) )^* = \{ y | Ay \in Q \} = A^{-1}(Q)$.