I'm reading a paper. In the proof of theorem 2, the authors refer to the Lipschitzness of Moreau-Yosida regularisation with citation. Like this:
For convex function $f$, Moreau-Yosida regularisation $g_{\eta}$ is defined as follows, $$ g_{\eta} = \inf_{y \in K} \left( \frac{1}{2 \eta} ||x-y ||^2 + f(y) \right) $$ Then, $g_{\eta}$ is Lipschitz continuous with constant $\frac{1}{2 \eta}$.
However, I cannot find such contents from the cited reference. Is it true?
EDIT: I think it's false cuz a representative example of Moreau-Yosida regularisation is Huber function $g_{\eta}(x) = \frac{1}{2\eta} x^2$ if $|x| \leq \eta$; otherwise, $|x| - \frac{\eta}{2}$, for $f(x) = |x|$.
Then, Is it true that Moreau-Yosida regularisation is Lipschitz continuous (on a compact set)?
EDIT2: I think that on a compact set $X$, the regularisation provides Lipschitz continuous function. Note that the cited theorem of the reference, Thm. 4.1.4, says that the gradient $\nabla g_{\eta}$ of $g_{\eta}$ is $\frac{1}{\eta} (x - prox(x))$ where $prox(x)$ is the proximal point. Hence, if the function is defined on a compact set, subset of real space, then the Lipschitz constant of $g_{\eta}$ will be given as $\frac{1}{\eta} \text{diam}(X)$ where $\text{diam}(X)$ is diameter of bounded set $X$.