Let $A, B \subseteq \mathbb{R}^3$ be convex sets. More precisely, $A$ is a convex cone, pointed in $0$, while $B$ is a ball with centre in $0$ and finite radius $R$. Clearly, $A\cap B$ is non-empty. If $\mathcal{H}=\{x\in\mathbb{R}^3 |\,r^Tx=0\}$ is a given hyperplane (viz. $r$ is known), is the following equality true?
$\Pi_{\mathcal{H}}(A)\cap \Pi_{\mathcal{H}}(B)=\Pi_{\mathcal{H}}(A \cap B)$
with $\Pi_{\mathcal{H}}(*)$ being the set containing the projection of all the points of the set $*$ onto $\mathcal{H}$.
The answer is generally negative. Here is a counterexample in the plane:
Let $A= \{ (t,t) | t \ge 0 \}$, $B= B(0,1)$ and $r=(0,1)$. Then, the projection of the intersection is $[0,{1 \over \sqrt{2}}] \times \{0\}$, but the intersection of the projections is $[0,1] \times \{0\}$.