Fix $a \in \mathbb R^n$ and let $\|\cdot\|$ be any norm on $\mathbb R$ (e.g $\ell\_1$ norm). For any $a \in \mathbb R^n$, it is clear that the function $f\_a(x) := \|x-a\|\_2 + \|x\|$ is strictly convex and has a unique minimizer $x(a)$.


**Question.** Given $a,b \in \mathbb R^n$, can $\|x(a)-x(b)\|\_2$ be bounded in terms of some norm of $a-b$ ?



Is the mapping $F(a):= \arg\min_{x \in \mathbb R^n} \|x-a\|_2 + \|x\|_1$ non-expansive?
First of all, the function $f\_a$ is not strictly convex, and hence, you should expect multiple minimizers. As such, non-expansiveness (even in some generalized sense) does not seem likely. Consider the one-dimensional case where
$$f\_a(x) = |x-a| + |x|$$
for which $\operatorname{argmin} f\_a = [0,a]$. So you can select minimizers which depend continuously on $a$, but this may not be what you want…


