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 $a-b$ ?