I am really struggling with this problem:
$C$ and $D$ are closed, convex subsets of ${R}^n$ with non-empty intersection, i.e. $C \cap D \neq \emptyset $ . Is it true that projection $p_{C\cap D}(x) = p_{C}(p_{D}(x))$ for any $x$? Any example of this or counter example?
Reminder: Projection is defined as: $p_{C}(x) = infimum\{\parallel x-y\parallel, y \in C \}$.
Hint: Consider two lines $C$ and $D$ through the origin in $\mathbb{R}^2$ and suppose that they meet in an angle $0 \lt \alpha \lt \pi/2$.
Observe:
The only points $x \in \mathbb{R}^2$ for which $p_C p_D(x) = 0$ are the points $x$ on the line $D^{\perp}$ through the origin which is perpendicular to $D$.
$p_C \circ p_D \neq p_D \circ p_C$, so the formula you ask about cannot hold since the roles of $C$ and $D$ in $p_{C \cap D}$ are symmetric.
Your definition of the projection $p_C$ in the reminder is not quite correct: the projection $x_C = p_C(x)$ is the unique point $x_C \in C$ for which $d(x_C,x) = \inf_{c \in C} \lVert c - x\rVert$.