Sup-Convolution of a Function on $\mathbb{R}^M$

Let $M$ be a natural number with $1\le M$ and let $\mathbb{R}$ be the real numbers, a Dedekind complete ordered field, with the order $\le$, the strict order $<$ and the quotient notation $s/t$ fixed there; write $2$ for the real number $1+1$, which satisfies $0<2$ by claim 8 of Elementary Order Arithmetic in an Ordered Field, so that $s/2$ is defined. For a real number $s$ write $s^{2}$ for the power $s\cdot s$.

Regard Euclidean space $\mathbb{R}^{M}$ as a real vector space, with the sum of points and the scalar multiple, write $x-\xi$ for the difference of points, and let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on $\mathbb{R}^{M}$.

Let $v:\mathbb{R}^{M}\to\mathbb{R}$ be a function whose set of values $\{v(x):x\in\mathbb{R}^{M}\}$ has an upper bound in $\mathbb{R}$, and let $\lambda\in\mathbb{R}$ satisfy $0<\lambda$. For $\xi\in\mathbb{R}^{M}$ put

$$S_{\lambda,v}(\xi)=\Bigl\{\,v(x)-\frac{\lambda}{2}\,\lVert x-\xi\rVert^{2}\ :\ x\in\mathbb{R}^{M}\Bigr\}.$$

This set is nonempty, and every upper bound $C$ for the values of $v$ is an upper bound for it: for each $x\in\mathbb{R}^{M}$ the real number $\frac{\lambda}{2}\lVert x-\xi\rVert^{2}$ is nonnegative, by claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square together with claim 5 of Elementary Arithmetic in an Ordered Field and claim 1 of Zero Products and Elementary Identities in a Field, so that $v(x)-\frac{\lambda}{2}\lVert x-\xi\rVert^{2}\le v(x)\le C$. Since $\mathbb{R}$ is Dedekind complete, $S_{\lambda,v}(\xi)$ has a least upper bound in $\mathbb{R}$.

The sup-convolution of $v$ with parameter $\lambda$ is the function $v^{\lambda}:\mathbb{R}^{M}\to\mathbb{R}$ whose value at $\xi$ is the least upper bound of $S_{\lambda,v}(\xi)$, written

$$v^{\lambda}(\xi)=\sup_{x\in\mathbb{R}^{M}}\Bigl(v(x)-\frac{\lambda}{2}\,\lVert x-\xi\rVert^{2}\Bigr).$$

In the symbol $v^{\lambda}$ the superscript is a label, not an exponent.