I am stuck with what looks like a basic problem from convex analysis and I don't seem to have the answer. Any pointers or insights would be of great help.
Suppose $K$ is a closed convex set in $\mathbb{R}^n$ and $R \in \mathbb{R}^{m \times n}$ is a matrix with $m<n$. Let $\tilde{K} = RK = \{y| \exists x \in K, y=Rx\}$, the projection of $K$ under $R$. Then $\tilde{K}$ is also closed and convex.
Let $T(x;C)$ be the tangent cone at $x \in C$ to a set $C$. It is easy to see that $RT(x;K) \subseteq T(Rx;RK)$ for any $x \in K$. I wanted to know if this can be strengthened somehow? Since both $T(Rx;Kx)$ and $RT(x;K)$ are cones, it seems plausible that $T(Rx;RK) = RT(x;K) +$ "something". If so, what is that something? Can more be said if $K$ were a polyhedron?