Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 339 / 3 / c / ii / Solution

The functionis the Moreau envelope of the convex indicator $I_D$, and is therefore convex. It is nonnegative and vanishes exactly on $D$. Since $C\cap D\ne \varnothing$, the minimum of $d_D$ over $C$ is zero, and every minimizer belongs to both $C$ and $D$.