Exact minimization of the linearly perturbed distance of a point to a closed convex set

Let $a, b \in \mathbb R^n$, and $C$ be a nonempty closed convex subset of $\mathbb{R}^n$ with $a \not \in C$. Consider the problem $$\text{minimize }\|x-a\| + b^Tx\text{ subject to } x \in C.$$ The case $b = 0$ corresponds to the well-known problem of projecting the point $a$ onto $C$. This problem can be easy or difficult, depending on the geometry of $C$ (for example, the latter problem is rather easy if $C$ is a diamond-shaped polyhedron, or an oval). However the perturbative case $b \ne 0$ is completely non-trivial and must be treated rather carefully. That notwithstanding, we show here that the case $\|b\| < 1$ is essentially equivalennt to the non-perturbative case $b = 0$.