Projection onto a dilated set

Question

Let $C \subset \mathbb{R}^n$ be non-empty, closed and convex. Also let $\delta>0$ and consider the dilated set defined as follows $$ D_{\delta}(C) = \{ x \in \mathbb{R}^n: \| x - \pi_{C}(x)\|_2 \le \delta\}, $$ where $\pi_C:\mathbb{R}^n \rightarrow C$ is the orthogonal projection onto $C$, which is well-defined since $C$ is non-empty, closed and convex. Assuming that I know how to calculate $\pi_C(x)$ for all $x \in \mathbb{R}^n$, I would like to know if it is possible to express $\pi_{D_\delta}$ in terms of $\pi_C$.

Answer 1

Let's call the dilated set $D$ for simplicity.
Let $x\in \mathbb{R}^n$.
If $x\in D$, then $\pi_D(x)=x$.
So assume $x\notin D$. Then the desired projection is $$\pi_D(x)=\pi_C(x)+\delta\frac{x-\pi_C(x)}{\|x-\pi_C(x)\|}.$$