Existence and Uniqueness of the Proximal Minimiser of a Convex Function

We work in the setting of Euclidean Space and Lebesgue Measure: Standing Notation, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the real numbers, and the Euclidean norm $\lVert\,\cdot\,\rVert$, dot product, distance $d_{E}$, topology, the notions of open, closed, bounded and compact subsets, and the closed balls $\bar{B}(x,r)$, 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, and let $x\in\mathbb{R}^{n}$. Define $\phi_{x}:\mathbb{R}^{n}\to\mathbb{R}$ by

Then the following holds.

1. (A unique minimiser) ¶ There is exactly one $y\in\mathbb{R}^{n}$ such that $\phi_{x}(y)\le\phi_{x}(z)$ for every $z\in\mathbb{R}^{n}$.