Properties of the Proximal Map of a Convex Function

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the real numbers, the Euclidean norm $\lVert\,\cdot\,\rVert$, dot product, distance $d_{E}$ and notion of openness, and the convention that Lipschitz maps between subsets of Euclidean spaces are understood for the restricted Euclidean distances, are as fixed there.

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be convex on $\mathbb{R}^{n}$, which is a convex subset of itself, let $\partial f=\partial_{\mathbb{R}^{n}}f$ be its subdifferential, and let $J=J_{f}$ be its proximal map. Then the following hold.

1. (Description of the fibres) (i) For all $x,y\in\mathbb{R}^{n}$ one has $J(x)=y$ if and only if $x-y\in\partial f(y)$.

2. (Surjectivity) (ii) For every $y\in\mathbb{R}^{n}$ and every $q\in\partial f(y)$ one has $J(y+q)=y$. In particular $J$ maps $\mathbb{R}^{n}$ onto $\mathbb{R}^{n}$.

3. (Firm nonexpansiveness) (iii) For all $x,x'\in\mathbb{R}^{n}$,

$$\lVert J(x)-J(x')\rVert^{2}\le\bigl(J(x)-J(x')\bigr)\cdot(x-x').$$

4. (Nonexpansiveness) (iv) For all $x,x'\in\mathbb{R}^{n}$ one has $\lVert J(x)-J(x')\rVert\le\lVert x-x'\rVert$; that is, $J$ is Lipschitz with constant $1$ on $\mathbb{R}^{n}$.

5. (The derivative where it exists) (v) Let $x\in\mathbb{R}^{n}$ be a point at which $J$ is differentiable, with derivative matrix the real matrix $A$ with $n$ rows and $n$ columns. Then

$$\lVert Ah\rVert^{2}\le(Ah)\cdot h\qquad\text{for every }h\in\mathbb{R}^{n}.$$

Moreover, if there is no $\nu\in\mathbb{R}^{n}$ with $\lVert\nu\rVert=1$ such that $\nu\cdot(Ah)=0$ for every $h\in\mathbb{R}^{n}$, then the map $h\mapsto Ah$ is injective.