There is an important proberty of subdifferentals of proper convex lsc functions, that $$\partial (f+g)(x)=\partial f(x)+\partial g(x)$$ holds as equality only under certain assumptions (see Rockafellar, Convex analysis, Th 23.8). What would be the counterexample of two such functions which fail this equality in general situation?
Especially, what would be the simplest example of two nonemply closed convex sets $C_1,C_2$ in $R^n$, such that $$N_{C_1}(x)+N_{C_2}(x) \subsetneq N_{C_1\cap C_2}(x)$$ where $N$ is a normal cone, i.e the subdifferental of a characteristic function? Is it possible for polyhedral sets?
EDIT: for polyhedral sets it is enough to have a nonempty intersection of the sets, not necessarily their interiors, see the same th in Rockafellar
Define $f$ by $$ f(x) =\begin{cases} -\sqrt x & x \ge 0\\ +\infty & x<0. \end{cases} $$ Then it holds $\partial f(0)=\emptyset$. Define $g(x)=f(-x)$. The subdifferential of $f+g$ at $x=0$ is equal to $\mathbb R$, while $\partial f(0)=\partial g(0)=\emptyset$.
For the second part of the question, take $C_1$ and $C_2$ closed balls that touch at exactly one point $x$. Then $N_{C_1\cap C_2}(x)$ is the whole space, but $N_{C_1}(x) + N_{C_2}(x)$ is one-dimensional.