Localisation of an Upper Semicontinuous Function by Projection onto a Closed Ball

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 absolute value $|\cdot|$ and least upper bounds, and the Euclidean norm $\lVert\,\cdot\,\rVert$, dot product, sum and difference of points, scalar multiples, Euclidean distance $d_{E}$, the origin $0_{\mathbb{R}^{n}}$, closed balls $\bar{B}(x,\rho)$, and the notions of openness, closedness, boundedness and compactness, are as fixed there; we abbreviate $\lVert z\rVert^{2}=\lVert z\rVert\cdot\lVert z\rVert$. For $s\in\mathbb{R}$ we write $s^{+}=s$ if $0\le s$ and $s^{+}=0$ otherwise, so that $0\le s^{+}$ and $s\le s^{+}$.

Let $r\in\mathbb{R}$ be positive and put $\bar{B}_{r}=\bar{B}(0_{\mathbb{R}^{n}},r)$. This set contains $0_{\mathbb{R}^{n}}$ and is bounded and closed by claims 1, 2 and 3 of Elementary Properties of the Closed Ball in a Metric Space, is convex by claim 2 of Euclidean Balls are Convex, and is compact by The Heine-Borel Theorem in $\mathbb{R}^n$, and Compactness of Closed Balls and the Unit Sphere; moreover $\bar{B}_{r}=\{\xi\in\mathbb{R}^{n}:\lVert\xi\rVert\le r\}$, since $d_{E}(0_{\mathbb{R}^{n}},\xi)=\lVert\xi\rVert$ by claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. Let

$$\pi_{r}:\mathbb{R}^{n}\to\mathbb{R}^{n}$$

send $\xi$ to the unique nearest point of $\bar{B}_{r}$ to $\xi$, which exists by claim 1 of Nearest-Point Projection onto a Nonempty Closed Convex Subset of Euclidean Space applied to the nonempty closed convex set $\bar{B}_{r}$. Continuity of maps between metric spaces and upper semicontinuity of real-valued functions are always understood with respect to $d_{E}$ on subsets of Euclidean spaces. Then the following hold.

1. (Description of the projection) (i) For every $\xi\in\mathbb{R}^{n}$ one has $\lVert\pi_{r}(\xi)\rVert\le r$; moreover $\pi_{r}(\xi)=\xi$ if and only if $\lVert\xi\rVert\le r$, and if $r<\lVert\xi\rVert$ then $\pi_{r}(\xi)=\bigl(r\,\lVert\xi\rVert^{-1}\bigr)\xi$. In all cases

$$\lVert\pi_{r}(\xi)-\xi\rVert=\bigl(\lVert\xi\rVert-r\bigr)^{+},\qquad \lVert\pi_{r}(\xi)+\xi\rVert\le\lVert\xi\rVert+r .$$

Finally $\pi_{r}$ is continuous on $\mathbb{R}^{n}$.

2. (Two elementary identities) (ii) For every $\xi\in\mathbb{R}^{n}$,

$$\bigl(\lVert\xi\rVert-r\bigr)^{+}\bigl(\lVert\xi\rVert+r\bigr)=\bigl(\lVert\xi\rVert^{2}-r^{2}\bigr)^{+}\qquad\text{and}\qquad \lVert\xi\rVert+r\le\bigl(\lVert\xi\rVert-r\bigr)^{+}+2r,$$

where $r^{2}=r\cdot r$ and $2r=r+r$.

3. (The localised function) (iii) Let $\Omega\subseteq\mathbb{R}^{n}$ be open with $\bar{B}_{r}\subseteq\Omega$, let $u:\Omega\to\mathbb{R}$ be upper semicontinuous on $\Omega$, and let $c\in\mathbb{R}$ satisfy $0\le c$. Let $v:\mathbb{R}^{n}\to\mathbb{R}$ be given by

$$v(\xi)=u\bigl(\pi_{r}(\xi)\bigr)-c\,\bigl(\lVert\xi\rVert^{2}-r^{2}\bigr)^{+}.$$

Then $v$ is upper semicontinuous on $\mathbb{R}^{n}$, the set of values of $v$ has an upper bound in $\mathbb{R}$, and $v(\xi)=u(\xi)$ for every $\xi\in\mathbb{R}^{n}$ with $\lVert\xi\rVert\le r$.