It is known that the characterization of projection onto a convex set $C$ is given by: $y=P_C(x)$ iff $\langle x-y, z-y\rangle \leq 0$ for any $z\in C$. Can we extend this conclusion to the neighborhood of $y$?
To be specific, let $C$ be a convex set, and $y, z\in \partial C$, such that the straight line between $y$ and $z$ is also on the boundary of $C$. Furthermore, assume that $C$ is symmetric, i.e. $x\in C$ implies $-x\in C$. Then, can we find a neighborhood of $y$ such that for any $x$ in that neighborhood, and any $v$ in the normal cone of $C$ at $x$, we have $\langle v, z-y\rangle \leq 0$?
If it is true, how to prove it? If not, any counterexamples?