Proximal / Prox Operator of a Unit Ball Function

Question

I have to calculate the proximal operator of the function $f(x)=δ$$_α$$_B$$(x)$ where B is the unit ball and α > 0.

I am very new to proximal mappings and would appreciate any help with how to solve this problem.

Answer 1

Fact: The prox of the indicator of a closed convex set is the euclidean projection operator thereupon.

Proof. Let $B$ be closed nonempty convex subset of a Hilbert space $\mathcal H$ (for example, $B$ be could be the closed ball for a given norm on $\mathcal H$), and let $\lambda > 0$. Then for any $x \in \mathcal H$, we have $$\text{prox}_{\lambda i_B}(x) := \text{argmin}_{y \in \mathcal H}\frac{1}{2}\|y-x\|_2^2 + \lambda i_B(y) = \text{argmin}_{y \in B}\frac{1}{2}\|y-x\|_2,$$ which is precisely the euclidean projection operator onto $B$. $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \Box$