The <squared distance to a convex set> satisfies
$$
\boxed{\nabla d_D(x)=x-P_Dx.}
$$
The map $I-P_D$ is firmly nonexpansive because $P_D$ is firmly nonexpansive, so
$$
\|\nabla d_D(x)-\nabla d_D(y)\|\leq\|x-y\|.
$$
Thus $d_D$ is one-smooth in the Euclidean norm.

Solution

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

{title2=$d_C(x)=\frac12\operatorname{dist}(x,C)^2$}

For a nonempty closed convex set $C$, the squared-distance function is the <Moreau envelope> of its <indicator functional of a constraint set>. It is convex and differentiable with
$$
\nabla d_C(x)=x-P_Cx,
$$
and this gradient is one-Lipschitz.

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