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Set $g(u)=tf(u)+\Vert u{\Vert }^2/2$. Expanding the square givesThe function $g$ is one-strongly convex, so its Fenchel conjugate is differentiable with one-Lipschitz gradient. The displayed identity therefore proves that the Moreau envelope $M_{t}f$ is differentiable even when $f$ is nonsmooth.

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 $D$.

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 withand this gradient is one-Lipschitz.