Find the polar cone of $K=\{y=(y_1,..y_m)\in R^{mn}|y_1+...+ y_m\in C\} $, where $C$ is a closed convex cone in $R^n$
I have started by using the Theorem of preimage that states: Assume that $C \in R^m$ is a closed convex cone and A is a $n$ x $m$ matrix. Let $K=\{x\in R^n|Ax \in C\}$. Then $K^o = \{A^T\lambda |\lambda \in C^o\}$
For this problem, $K^o=\{\lambda= (\lambda_1,...,\lambda_1)\in R^n| \lambda \in C^o\}$.
I am not sure how to finish this problem...mostly rewriting the $\lambda\in C^o$ part.