It is well-known that
If $C$ is a closed and convex set in $\mathbb R^n$. Then $||\text{proj}_C(x) - \text{proj}_{C}(y)||\leq ||x-y||$ for all $x, y\in \mathbb R^n$.
Here $\text{proj}_C(x) =\arg\min_{z\in C} ||x-z||$ is the closest-point projection of $y$ onto closed convex set $C$.
I am wondering that
If $A, B$ are closed and convex sets in $\mathbb R^n$. Then do we have a "tight" upper bound for $||\text{proj}_A(v) - \text{proj}_{B}(v)||$ where $v$ is some given point in $\mathbb R^n$.
My particular interested case is that $A\subset B$. Is there any ideas? Thanks!